\documentclass[11pt,reqno]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,amscd,mathtools,mathrsfs} \usepackage{booktabs,array,longtable,needspace} \usepackage{microtype} \usepackage[hidelinks]{hyperref} \hypersetup{pdftitle={A proof of the Hausel--Rodriguez-Villegas conjecture},pdfauthor={Qiyue Tang},pdfsubject={Mixed Hodge polynomials of primitive-twisted GLn character varieties}} \usepackage[margin=1.12in]{geometry} \allowdisplaybreaks[2] \emergencystretch=1.5em \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{convention}[theorem]{Convention} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\Q}{\mathbf{Q}} \newcommand{\C}{\mathbf{C}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\A}{\mathbf{A}} \newcommand{\Gm}{\mathbf{G}_m} \newcommand{\OO}{\mathcal{O}} \newcommand{\LL}{\mathbb{L}} \newcommand{\TT}{\mathbf{T}} \newcommand{\Th}{\vartheta} \newcommand{\La}{\Lambda} \newcommand{\IC}{\operatorname{IC}} \newcommand{\MHM}{\operatorname{MHM}} \newcommand{\MMHM}{\operatorname{MMHM}} \newcommand{\MHS}{\operatorname{MHS}} \newcommand{\MMHS}{\operatorname{MMHS}} \newcommand{\pH}{{}^{p}\!\mathcal{H}} \newcommand{\Gr}{\operatorname{Gr}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\RHom}{\operatorname{RHom}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ext}{\operatorname{Ext}} \newcommand{\Cone}{\operatorname{Cone}} \newcommand{\fib}{\operatorname{fib}} \newcommand{\cofib}{\operatorname{cofib}} \newcommand{\Crit}{\operatorname{Crit}} \newcommand{\Perf}{\operatorname{Perf}} \newcommand{\Pic}{\operatorname{Pic}} \newcommand{\Bun}{\operatorname{Bun}} \newcommand{\Quot}{\operatorname{Quot}} \newcommand{\Hilb}{\operatorname{Hilb}} \newcommand{\Conf}{\operatorname{Conf}} \newcommand{\GL}{\operatorname{GL}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\id}{\operatorname{id}} \newcommand{\supp}{\operatorname{supp}} \newcommand{\im}{\operatorname{im}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\wt}{\operatorname{wt}} \newcommand{\sgn}{\operatorname{sgn}} \newcommand{\Log}{\operatorname{Log}} \newcommand{\Exp}{\operatorname{Exp}} \newcommand{\pt}{\mathrm{pt}} \newcommand{\red}{\mathrm{red}} \newcommand{\vir}{\mathrm{vir}} \renewcommand{\ss}{\mathrm{ss}} \newcommand{\sslash}{\mathbin{/\mkern-6mu/}} \newcommand{\reg}{\mathrm{reg}} \newcommand{\nil}{\mathrm{nil}} \newcommand{\DR}{\mathrm{DR}} \newcommand{\cS}{\mathscr{S}} \newcommand{\cI}{\mathscr{I}} \newcommand{\cJ}{\mathscr{J}} \newcommand{\cP}{\mathscr{P}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cM}{\mathcal{M}} \newcommand{\eps}{\varepsilon} \newcommand{\into}{\hookrightarrow} \newcommand{\xto}[1]{\xrightarrow{#1}} \newcommand{\bbone}{\Q} \newcommand{\abs}[1]{\lvert #1\rvert} \newcommand{\norm}[1]{\lVert #1\rVert} \title[A proof of the HRV conjecture]{A proof of the Hausel--Rodriguez-Villegas conjecture} \author{Qiyue Tang} \address{Zhili College, Tsinghua University, Beijing 100084, China} \email{tangqy24@mails.tsinghua.edu.cn} \date{November 13, 2026} \subjclass[2020]{14D20, 14C30, 14N35, 14F45} \keywords{Character varieties, mixed Hodge structures, Higgs bundles, stable pairs, perverse filtration} \begin{document} \begin{abstract} We prove the Hausel--Rodriguez-Villegas conjecture on the mixed Hodge polynomials of primitive-twisted $\GL_n$ character varieties. \end{abstract} \maketitle \tableofcontents \section{Introduction} \label{sec:introduction} Let $C$ be a smooth projective connected complex curve of genus $g\geq0$, and write $K_C$ for its canonical line bundle. For a positive integer $n$ and a primitive $n$th root of unity $\zeta$, set \begin{equation}\label{eq:character-variety} \cM_{g,n}= \left\{(A_1,B_1,\ldots,A_g,B_g)\in\GL_n(\C)^{2g}: \prod_{j=1}^{g}[A_j,B_j]=\zeta I_n\right\}\sslash\GL_n(\C). \end{equation} Here $[A,B]=ABA^{-1}B^{-1}$ and $\sslash$ denotes the affine geometric invariant theory quotient for simultaneous conjugation. Put $N_n=(g-1)n^2+1$. For $g\geq1$, this twisted character variety is smooth of complex dimension $2N_n$. Its cohomology is of Hodge--Tate type \cite[Corollary 4.1.11]{HRV}. We write $W$ for Deligne's weight filtration, use ordinary cohomology, and set \begin{equation}\label{eq:mixed-polynomial} H(\cM_{g,n};q,t)= \sum_{D,k}\dim\Gr^W_{2k}H^D(\cM_{g,n},\Q)\,q^kt^D. \end{equation} For a box $b$ of a partition $\lambda$, let $a_b$ and $l_b$ be its arm and leg lengths, counting boxes to its right and below, respectively, and put $h_b=a_b+l_b+1$. With $Q$ a formal rank variable, define \begin{align} \mathcal H_{\lambda}(z,w) &=\prod_{b\in\lambda} \frac{(z^{2a_b+1}-w^{2l_b+1})^{2g}} {(z^{2a_b+2}-w^{2l_b})(z^{2a_b}-w^{2l_b+2})},\label{eq:hook-function}\\ \Omega(Q;z,w)&=\sum_{\lambda}Q^{|\lambda|}\mathcal H_{\lambda}(z,w),\label{eq:omega}\\ \overline H_n(z,w)&=(z^2-1)(1-w^2)[Q^n]\Log\Omega(Q;z,w). \label{eq:hrv-function} \end{align} The empty partition contributes $1$. Here $\mu$ denotes the M\"obius function and $\Log$ is the plethystic logarithm for the Adams operations $\psi_d(Q,z,w)=(Q^d,z^d,w^d)$; explicitly, \begin{equation}\label{eq:plethystic-log} \Log F=\sum_{d\geq1}\frac{\mu(d)}d \log F(Q^d;z^d,w^d). \end{equation} \begin{theorem}\label{thm:hrv} For every $g\geq2$ and every $n\geq1$, \begin{equation}\label{eq:hrv-main} H(\cM_{g,n};q,t) =(t\sqrt q)^{2N_n} \overline H_n\!\left(\sqrt q,-\frac1{t\sqrt q}\right). \end{equation} The expression on the right is independent of the choice of the square root and is a polynomial in $q,t$. \end{theorem} \begin{corollary}\label{cor:hrv-all-genera} For every $g\geq0$ and every $n\geq1$, the mixed Hodge polynomial of $\cM_{g,n}$ is given by \eqref{eq:hrv-main}. \end{corollary} \begin{proof} Theorem~\ref{thm:hrv} treats $g\geq2$. If $g=0$, the empty commutator product is $I_n$, so $\cM_{0,1}$ is a point and $\cM_{0,n}$ is empty for $n>1$. The genus-zero identity \cite[Theorem 4.3.1]{HRV} gives \[ \Log\Omega(Q;z,w)=\frac{Q}{(z^2-1)(1-w^2)}. \] Thus $\overline H_1=1$ and $\overline H_n=0$ for $n>1$. Since $N_1=0$, these are exactly the required mixed Hodge polynomials; the zero expression remains zero when $N_n<0$. If $g=1$, one has $\cM_{1,n}\cong(\C^*)^2$ \cite[Theorem 2.2.17]{HRV}. The genus-one identity \cite[Theorem 1.0.2]{CRV}, specialized at $u=1$, is \[ \Log\Omega(Q;z,w) =\frac{(z-w)^2Q}{(z^2-1)(1-w^2)(1-Q)}. \] Hence $\overline H_n(z,w)=(z-w)^2$ for all $n\geq1$. Here $N_n=1$, and substitution gives \[ (t\sqrt q)^2 \left(\sqrt q+\frac1{t\sqrt q}\right)^2 =(1+qt)^2=H((\C^*)^2;q,t). \] \end{proof} \subsection{Background and previous results} Character varieties and Higgs moduli provide two algebraic structures on the spaces arising from nonabelian Hodge theory. Hitchin introduced the moduli of solutions of the self-duality equations on a Riemann surface and its integrable system in \cite{HitchinSD,HitchinIS}. Simpson developed the correspondence between Higgs bundles and local systems in \cite{Simpson}. For $g\geq2$, in the primitive-twisted case the Betti space \eqref{eq:character-variety} is diffeomorphic to the moduli of stable $K_C$-Higgs bundles of rank $n$ and degree $d$, where $\gcd(n,d)=1$ and $\zeta=\exp(2\pi\mathrm i d/n)$. The Hitchin map sends a Higgs bundle to the coefficients of its characteristic polynomial. It is proper, and its generic fiber is a Picard variety of fixed degree on a smooth spectral curve. Its perverse Leray filtration provides a second grading on Higgs cohomology. On the Betti side, Deligne's weight filtration records the mixed Hodge structure of the affine character variety. Hausel and Thaddeus related mirror symmetry for the $\operatorname{SL}_n$ and $\operatorname{PGL}_n$ Hitchin systems to stringy Hodge invariants \cite{HT}. The mixed Hodge partition formula was announced in Hausel's 2004 survey in the equivalent $\operatorname{PGL}_n$ normalization \cite[Conjecture 5.1]{HauselSurvey}. Hausel and Rodriguez-Villegas developed it in their 2006 preprint, published in 2008 \cite[Conjecture 4.2.1]{HRV}. Their starting point was arithmetic: the character table of $\GL_n(\mathbb F_q)$ gives point counts, and Katz's theorem in the appendix converts polynomial counts into compactly supported $E$-polynomials. The resulting partition sum suggested the bigraded mixed Hodge formula. The two gradings in \eqref{eq:mixed-polynomial} retain both cohomological degree and weight. With compactly supported $E$-polynomials, Poincar\'e duality gives \[ E_c(\cM_{g,n};q)=q^{2N_n}H(\cM_{g,n};q^{-1},-1). \] Their paper also established the rank-two formula in arbitrary genus and related the predicted symmetries to curious hard Lefschetz and quiver invariants \cite[Theorem 1.1.3 and Sections 4--5]{HRV}. Hausel, Letellier and Rodriguez-Villegas subsequently formulated a Macdonald-polynomial extension for generic punctured character varieties and proved its arithmetic and quiver-theoretic specializations \cite{HLRV}. The small-genus cases have explicit partition identities. For genus zero the primitive-twisted variety is a point in rank one and is empty in higher rank; the corresponding identity is \cite[Theorem 4.3.1]{HRV}. For genus one it is $(\mathbb C^*)^2$ in every rank. Carlsson and Rodriguez-Villegas proved the required identity using vertex operators \cite[Theorem 1.0.2]{CRV}; it also follows from the Macdonald-polynomial Nekrasov--Okounkov formula of Rains and Warnaar \cite[Theorem 1.3]{RW}. In the notation above, $\overline H_n(z,w)=(z-w)^2$ for $g=1$ and every $n\geq1$. For any genus, the rank-one variety is $(\mathbb C^*)^{2g}$ and its mixed Hodge polynomial is $(1+qt)^{2g}$. Theorem~\ref{thm:hrv} treats all ranks in genus at least two for the primitive central twist in \eqref{eq:character-variety}. Several general specializations were proved through arithmetic and symmetric functions. Schiffmann counted indecomposable vector bundles and stable Higgs bundles over finite fields \cite{Schiffmann}. Mozgovoy and Schiffmann obtained formulas for twisted Higgs bundles and type-$A$ quiver bundles \cite{MozSchiff}. Mellit proved the predicted Poincar\'e polynomial for the unpunctured coprime moduli \cite{MellitHiggs}, and established the punctured Poincar\'e formula through Macdonald polynomials and affine Springer fibers \cite{MellitPoincare}. His integrality theorem for the Hausel--Letellier--Villegas kernels controls the coefficients of the symmetric-function series \cite{MellitIntegral}. His toric stratifications establish curious hard Lefschetz for the character varieties \cite{MellitToric}. The distinction among the counting, integrality, Poincar\'e, and full mixed Hodge statements is recorded in \cite[Conjecture 4.5 and Theorems 4.6--4.8]{MellitSurvey}. The relation between the two geometric gradings was formulated by de Cataldo, Hausel and Migliorini as the $P=W$ conjecture. They proved it in rank two for the groups $\GL_2$, $\operatorname{SL}_2$, and $\operatorname{PGL}_2$ \cite{dCHM}. For coprime $\GL_n$ moduli in arbitrary rank, the theorem was proved by Maulik and Shen \cite{MSPW} and by Hausel, Mellit, Minets and Schiffmann \cite{HMMS}. Maulik, Shen and Yin obtained a further proof through perverse filtrations and Fourier transforms \cite{MSY}. With the normalized perverse filtration, $P=W$ identifies $P_k$ with $W_{2k}=W_{2k+1}$ under nonabelian Hodge theory. The remaining enumerative task is to calculate the dimensions in each cohomological degree and each perverse degree of the Hitchin system. The connection with curve counting predates these proofs of $P=W$. Chuang, Diaconescu and Pan expressed the HRV partition sum as a refined Gopakumar--Vafa expansion for a local Calabi--Yau curve and identified its BPS multiplicities with the primitive multiplicities in the relative Lefschetz decomposition of perverse graded Higgs cohomology \cite{CDP}. Their construction used a proposed refined stable-pair formula together with $P=W$. Chuang, Diaconescu, Donagi and Pantev extended this approach to parabolic refined invariants and Macdonald polynomials \cite{CDDP}. These works supply the enumerative origin of the local-curve series used here. Monavari described the fixed stable-pair loci by double nested Hilbert schemes and proved their Euler-characteristic hook product \cite{MonavariNested}. He subsequently computed the $K$-theoretic local Donaldson--Thomas series in arbitrary genus and degree and proved the local $K$-theoretic DT/PT correspondence \cite{MonavariRefined}. Here we calculate the cohomological stable-pair series in mixed Hodge modules, retaining the orientation maps, finite symmetric actions, and cohomological grading. \subsection{Proof of the formula} We compute cohomological stable-pair invariants on $X_1=\Tot_C(K_C\oplus\OO_C)$ in two ways. A positive quotient $K_C\oplus\OO_C\twoheadrightarrow L$ relates $X_1$ to the split threefold $X_0=\Tot_C(L\oplus K_CL^{-1})$. Both are described by critical charts on the derived stack of surface pairs on $\Tot_C L$. Section~\ref{sec:physical} constructs these charts and compares their closed forms and orientations. Sections~\ref{sec:comparison}--\ref{sec:hecke} construct the comparison morphism over the positive Hitchin base. Elementary modifications act on this morphism and its cofiber. Their commutator is multiplication by the rank. After passing to the quotient by contributions with at least two smaller positive ranks, the cofiber is supported in a finite interval of Euler characteristics. The commutator forces it to vanish. Section~\ref{sec:source} computes the positive source using the Hitchin support theorem, the smooth-stack integrality theorem, and the relative Hilbert schemes of reduced spectral curves. A critical calculation identifies its primitive contribution with the individual perverse direct-image coefficients of the canonical Hitchin map. Disjoint spectral packets and induction on rank then give the global source identity. Its proof uses the comparison morphism and its orientation constructed in the preceding sections. Section~\ref{sec:fock} computes the same stable-pair series by two torus actions. The fixed loci are Quot schemes of the dualizing modules of monomial thickenings of $C$. The full attracting filtrations give purity in every cohomological degree. Their indices and finite permutation actions express the series through symmetric powers of $C$ and the arms, legs, and hooks of partitions. Throughout these calculations, $H^1(C,\Q)$ remains a Hodge structure. In Section~\ref{sec:recovery}, purity separates the cohomological degrees in the equality of Hodge classes, and $P=W$ identifies the resulting dimensions with the coefficients of \eqref{eq:mixed-polynomial}. Taking the weight character yields \eqref{eq:hrv-main}. \subsection{Notation and coefficients} For the proof of Theorem~\ref{thm:hrv}, assume $g\geq2$ and put $h=g-1$. Choose a maximal rank for the argument; every construction below is coefficientwise in this finite rank range. Choose a line bundle $L$ of sufficiently large even degree $\ell$ and sections \begin{equation}\label{eq:positive-quotient} q_0=(a,b):\OO_C\oplus K_C\twoheadrightarrow L, \qquad A=K_CL^{-1}. \end{equation} Such sections exist without common zero for $\ell$ sufficiently large. The extension class of $0\to A\to\OO_C\oplus K_C\to L\to0$ is $\alpha_q\in H^1(C,K_CL^{-2})$. The two threefolds \begin{equation}\label{eq:physical-rows} X_0=\Tot_C(L\oplus A),\qquad X_1=\Tot_C(K_C\oplus\OO_C) \end{equation} have the volume forms induced by their determinant identifications with $K_C$. Write \begin{equation}\label{eq:rank-constants} B_r=\ell r^2,\qquad a_r=-hr-\frac{\ell r(r-1)}2,\qquad \alpha_r=\frac{a_r}{r},\qquad c_r=\alpha_r-\frac1{2r},\qquad N_r=hr^2+1. \end{equation} The symbol $a_r$ in \eqref{eq:rank-constants} is unrelated to an arm length $a_b$. For a line bundle $D$ on $C$, set $\cA_r^D=\bigoplus_{j=1}^rH^0(C,D^j)$. The Hitchin map records the coefficients of the characteristic polynomial, using the same convention for all $D$. Multiplication of monic spectral polynomials defines the finite addition maps \begin{equation}\label{eq:spectral-addition} \operatorname{add}:\prod_{\nu}\cA_{r_\nu}^D\longrightarrow\cA_r^D, \qquad \sum_\nu r_\nu=r. \end{equation} Their finiteness will also be proved directly below. We use $\Exp_{\operatorname{add}}$ and $\Sym_{\operatorname{add}}$ for the symmetric-power operations using these maps, including their actual finite permutation projectors. We write $\MHS$ for rational mixed Hodge structures, $\MHM(Y)$ for rational algebraic mixed Hodge modules on $Y$, and $\MMHM(Y)$ for their monodromic version; put $\MMHS=\MMHM(\mathrm{pt})$. The symbol $t_0$ denotes classical truncation, and $\IC_Y$ is the perverse-normalized intersection complex. All six operations are derived. The notation $\omega_Y$ denotes the Hodge dualizing complex, $\mathbb D$ denotes Verdier duality, and $\pH^i$ denotes perverse cohomology. We use the perverse normalization of monodromic vanishing cycles $\phi_f$. When taking vanishing cycles and their products, we use the conventions of \cite{DM}. Set \begin{equation}\label{eq:tate-conventions} \TT=\Q(-1),\qquad \LL=\TT[-2],\qquad \Th^2=\TT,\qquad \La=\Th[-1]. \end{equation} For every integer $m$, write $\TT^{m/2}=\Th^m$ and $\LL^{m/2}=\La^m$. The tensor product in $\MMHM$ is the monodromic convolution product. The half-Tate object $\Th$ has even self-symmetry; $\La$ has odd self-symmetry. Thus $\La^2=\LL$. We prove the relevant permutation calculation in Section~\ref{sec:fock}. The intrinsic weights are $\wt(\Th)=1$ and $\wt(\La)=0$. For a pure complex $K$ of weight $w$, $\pH^i(K)$ is pure of weight $w+i$. The ordinary embedding $\MHM(Y)\to\MMHM(Y)$ is exact, fully faithful, and monoidal. It has an exact fiber realization to ordinary mixed Hodge modules on its image. We use this to return to ordinary coefficients only after proving that the objects in question belong to that image. All preceding products are formed in $\MMHM$. If $K$ has bounded finite-dimensional cohomology, $[K]$ denotes its alternating class in $K_0(\MHS)$ or $K_0(\MMHS)$, as appropriate. Square brackets around a complex, such as $[P_i[-i]]$, always retain its Hodge structure. For a smooth moduli space, $\Q[d]$ denotes its constant Hodge module normalized in complex dimension $d$. For an oriented critical moduli space $Y$, $\Phi_Y$ is its normalized monodromic Donaldson--Thomas coefficient. On a smooth zero-potential critical chart its value is the corresponding constant object with the normalization \eqref{eq:tate-conventions}. The construction and all orientation maps used in this paper are given in Section~\ref{sec:physical}. \subsection{Moduli and the two support categories} A compactly supported stable pair on $X_i$ is a map $\OO_{X_i}\to F$ with $F$ pure of dimension one and zero-dimensional cokernel. Write $P_{r,\chi}(X_i)$ for its moduli space, where the pushdown of $F$ to $C$ has rank $r$ and Euler characteristic $\chi$. The framed stack used to compare stability chambers is the derived stack of surface pairs on $S=\Tot_C(L)$, with its two additional dual fields in the critical model. Its ordinary surface data are \begin{equation}\label{eq:surface-data} (E,\Theta,s),\qquad \Theta:E\longrightarrow E\otimes L,\quad s:\OO_C\longrightarrow E. \end{equation} Here $E$ is a vector bundle. The additional fields, the relation variable, and the coherent realization of its framing are described in Section~\ref{sec:physical}. We use Euler slope $\mu(E)=\chi(E)/\rk(E)$ and the Harder--Narasimhan filtration of the $L$-Higgs pair $(E,\Theta)$. A cutoff $c$ means $\mu_{\min}(E,\Theta)>c$. The symbol $\cP_{r,\chi}(c)$ denotes the corresponding nonzero-section surface window. The potential is \begin{equation}\label{eq:quadratic-potential} g_q(E,\Theta)=\frac12 \langle\alpha_q,\Tr(\Theta^2)\rangle. \end{equation} It factors through a function $g_{\cA}$ on $\cA_r^L$. For a rank partition $\nu=(r_1,\ldots,r_s)$ of $r$, let $\Sigma_\nu\subset\cA_r^L$ be the image of spectral addition. There are two different Serre subcategories of perverse Hodge modules in the argument. The category $\cS_r$ consists of objects whose simple constituents have strict supports among the $\Sigma_\nu$. It includes the single rank-$r$ semistable unframed coefficient. The category $\cI_r$ is generated by the perverse constituents of finite-addition convolutions of the framed and unframed coefficients occurring in proper flag correspondences, with at least two positive-rank factors, all of rank less than $r$. We include their indicated shifts, Tate twists, and source vanishing-cycle images. In particular $\cI_1=0$. For either category $\cJ$ let $D_{\cJ}$ denote the complexes all of whose perverse cohomology objects belong to $\cJ$. Direct images from the intermediate stacks can be unbounded on one side. Statements about them are always made in a fixed perverse degree. Grothendieck classes are taken at the bounded proper endpoints. A finite-type smooth presentation gives a common lower standard bound on the ordinary pullbacks to its \v Cech nerve. The $m$th totalization fiber has this bound increased by $m$; therefore only finitely many levels contribute in any fixed degree. On the finite-dimensional Hitchin base, the same observation gives a finite Postnikov range for each fixed perverse degree of a finite convolution. All Grothendieck classes used later belong to bounded proper stable-pair or Hitchin endpoints. \subsection{Established inputs} For a sufficiently positive effective $L$, the positive Hitchin support theorem and cohomological $\chi$-independence hold in the derived category of mixed Hodge modules \cite[Theorems 0.1 and 0.4]{MSChi}. The smooth-stack integrality theorem of \cite[Theorem 10.2.7 and Lemma 10.2.9]{BDIKP}, with its commutative cotangent orientation, applies to the fixed-slope positive Higgs stack. Section~\ref{sec:source} verifies its hypotheses and fixes the scalar stabilizer parameter. The integral localization theorem of \cite[Corollary 7.19]{KPS} supplies the morphism for filtered oriented $3$-Calabi--Yau objects. We use its chart-level monodromic Hodge realization together with the specialization exchange of \cite{Descombes}. The smooth descent and the comparison of orientations are included below. The conductor transversality criterion of \cite[Theorem 4.16(3)]{MSV} supplies smoothness of the relative Hilbert schemes on the reduced positive spectral locus. Finally, smooth semiprojective Higgs moduli have pure cohomology \cite[Corollary 1.3.2]{HRVPurity}. The $P=W$ theorem for coprime $\GL_n$ Higgs moduli is \cite[Theorem 0.2]{MSPW}. The rational tautological classes have the Hodge types stated in \cite{Shende}; together with tautological generation, they identify the resulting weight pieces with Tate structures. These facts are used only after the two stable-pair calculations and their comparison have been proved. \section{Critical models and their orientations}\label{phys:section}\label{sec:physical} We first construct the comparison of coefficients used below. All determinants in this section are graded determinants. Products of determinants therefore include their Koszul commutativity maps. The derived category of perfect complexes is used with its stable enhancement, so that a fiber, a map of triangles, and their compatibilities refer to the corresponding enhanced constructions. \subsection{The framed critical model} Let \(N\) be a rank-two bundle on \(C\), with a fixed identification \(\det N=K_C\), and choose an exact sequence \begin{equation}\label{phys:extension} 0\longrightarrow K_CL^{-1}\longrightarrow N \xrightarrow{q}L\longrightarrow0. \end{equation} Its signed extension class is denoted by \(\alpha_q\in H^1(C,K_CL^{-2})\). The sign is fixed by writing a transition between local lifts of the quotient coordinate as \(X_j=X_i-c_{ij}\Theta\), with \(c_{ij}\) representing \(\alpha_q\). Put \(S=\Tot_C L\), with projection \(p:S\to C\). The map \(\Tot_C N\to S\) is the \(K_S\)-torsor with class \(\alpha_q y\), where \(y\) is the tautological section of the pullback of \(L\). Let \(\cP_L\) be the derived stack of surface pairs \(\OO_S\xrightarrow{s}G\), with \(G\) perfect and properly supported, and write \[ J_G=\fib(\OO_S\to G). \] We work on the locus on which the ordinary sheaf \(G\) is finite over \(C\). It can then be described by its curve pushdown \(E\), the endomorphism \(\Theta:E\to E\otimes L\), and its section. The regular function used on this stack is \begin{equation}\label{phys:potential} g_q(E,\Theta,s) =\frac12\,\Tr_C\!\left(\alpha_q\Tr(\Theta^2)\right). \end{equation} Here \(\Tr_C:H^1(C,K_C)\to\C\) is Serre trace. Formula \eqref{phys:potential} defines a function on the derived stack as well: use a Čech or Thom--Whitney representative of \(\alpha_q\) and the universal trace. Its derivative in a bundle and field direction \((b,v)\) is \(\Tr_C(\alpha_q\Tr(\Theta v))\), since the additional term \(\Tr(\Theta[b,\Theta])\) vanishes. The physical enhancement of a stable pair on \(X=\Tot_C N\) is the moduli of perfect complexes \[ I=\fib(\OO_Y\to F),\qquad I|_D\simeq\OO_D, \quad Y=\mathbf P_C(\OO_C\oplus N),\quad D=\mathbf P_C N. \] The boundary is framed by the indicated isomorphism. Whenever a determinant trivialization is also used, its restriction agrees with the determinant of this boundary framing, with the specified homotopy of agreement. On a formal neighborhood of this locus, boundary framing agrees with fixing the determinant: near \(\OO_D\), the perfect-object stack is the derived Picard stack, and \(R\Gamma(Y,\OO_Y)\to R\Gamma(D,\OO_D)\) is an equivalence. Thus restriction to \(X\), followed by extension with the fixed determinant, also identifies the formal problems for the two compactifications used below. The stable-pair locus is an open condition on the classical truncation. Pure one-dimensional sheaves in this locus have locally free curve pushdown: a curve-torsion submodule would generate a zero-dimensional submodule under the finite support algebra. \begin{proposition}\label{phys:critical-equivalence} The boundary-framed physical enhancement is naturally equivalent, on the indicated locus, to \(\Crit(\cP_L,g_q)\). The equivalence is defined over connective Artin differential graded algebras, and respects maps of objects, their homotopies, direct sums, and finite exact diagrams. \end{proposition} \begin{proof} Compactify the torsor over \(S\) as \(\pi:Z=\mathbf P_S(V)\to S\), using the lines convention, where \[ 0\longrightarrow K_S\longrightarrow V\longrightarrow\OO_S \longrightarrow0,\qquad H=\OO_Z(D_\infty). \] The projective bundle formulas give \begin{equation}\label{phys:p1-identities} R\pi_*H=V^\vee,\qquad R\pi_*H^{-1}=0,\qquad R\pi_*H^{-2}=K_S[-1]. \end{equation} The relative diagonal has resolution \[ 0\longrightarrow H_1^{-1}H_2^{-1}\det(V)^\vee \longrightarrow\OO_{Z\times_S Z} \longrightarrow\OO_{\Delta_Z}\longrightarrow0. \] For a framed perfect object \(I\), define \[ G=R\pi_*(IH^{-1})[1],\qquad U=R\pi_*(IH^{-2}K_S^{-1})[1],\qquad Q=GK_S^{-1}. \] Applying this diagonal resolution and then tensoring by \(H\) recovers \begin{equation}\label{phys:diagonal-reconstruct} I\simeq\fib(\pi^*U\longrightarrow\pi^*G\,H). \end{equation} Adjunction identifies the arrow with \(U\to GV^\vee\). Since \(H|_{D_\infty}=K_S^{-1}\), the framing identifies its boundary restriction with \(U\to Q\). Thus the framed object is equivalently a map of triangles \[ \begin{CD} \OO_S @>>> U @>>> Q @>>> \OO_S[1]\\ @V{s}VV @VVV @V{\id}VV @V{s[1]}VV\\ G @>>> GV^\vee @>>> Q @>{c_G}>> G[1]. \end{CD} \] If \(e:Q\to\OO_S[1]\) is the upper connecting map, the middle arrow includes the specified homotopy \(s[1]e\simeq c_G\). In cone coordinates, write \(\varphi\) for a degree-zero cochain in \(\RHom_S(Q,G)\), and view \(e\) as a degree-one cochain in \(\RHom_S(Q,\OO_S)\). The equations are \begin{equation}\label{phys:homotopy-equations} d\varphi+se=c_G,\qquad de=0. \end{equation} Conversely these equations construct the upper triangle, its map to the lower triangle, and the fiber \eqref{phys:diagonal-reconstruct}. Both composites are the unit and counit of the displayed diagonal kernel. This also proves the assertion on mapping spaces and higher homotopies, after every derived base change. The tangent of \(\cP_L\) at a pair is \(\RHom_S(J_G,G)\). Surface duality with proper support identifies its shifted cotangent with \(\RHom_S(GK_S^{-1},J_G[1])\). Consequently \eqref{phys:homotopy-equations} describes the derived zero stack of the cotangent section \((c_G,0)\). In the monic resolution \[ [\,p^*E\otimes L^{-1}\xrightarrow{y-\Theta}p^*E\,] \] multiplication by \(y\) is chain homotopic to multiplication by \(\Theta\). The torsor cocycle \(\alpha_q y\) therefore represents the covector \(\Tr_C(\alpha_q\Tr(\Theta\,d\Theta))\), including bundle directions by the trace identity preceding the proposition. It is \(dg_q\). In a local splitting of the quotient, $X:E\to E\otimes A$ is the transverse field, and $\eps:E\to K_C$ is the dual framing field. Write $\psi$ for the cotangent field dual to $\Theta$, and $\rho_{\mathrm{cot}},\rho_{\mathrm{phys}}$ for the relation generators in the cotangent and physical models. The cone convention in \eqref{phys:homotopy-equations} gives \begin{equation}\label{phys:dictionary} \psi=-X,\qquad e=-\eps,\qquad \rho_{\mathrm{cot}}=-\rho_{\mathrm{phys}}. \end{equation} The last sign follows from the first two: they negate the moment expression, so they also negate its degree-minus-one relation variable. This proves the claimed equivalence. \end{proof} For the split bundle \(L\oplus K_CL^{-1}\), the function is zero. For \(N=K_C\oplus\OO_C\) and the fixed positive quotient \(q_0\), it is the function \eqref{phys:potential}. These two uses have the same cone convention and the same dictionary \eqref{phys:dictionary}. \subsection{The full deformation complex and its trace} We record a model that determines both the closed form and its orientation. It is useful to allow a framing bundle \(R\) in place of \(\OO_C\). On a curve and bundle chart, write \(V_0\) for the curve pushdown and \(W\) for the two normal directions, with \(\Delta=\det W\). The universal coefficient algebra has ordinary coordinates \(\Theta,X,s,\eps\), and all matrix entries of an odd coordinate \(\rho\), with \[ |\rho|=-1,\qquad d\rho=[\Theta,X]+s\eps. \] For homogeneous coefficients and homogeneous matrix maps we use \((a\otimes f)(b\otimes h)=(-1)^{|f||b|}ab\otimes fh\). On the relative projective plane, let \([z_0:z_1:z_2]\) be the homogeneous coordinates, with \(D=(z_0=0)\). The universal perfect complex has terms \[ M^{-1}=V_0\Delta^{-1}\OO(-2),\quad M^0=(V_0\otimes W^\vee)\OO(-1)\oplus R,\quad M^1=V_0\OO \] and arrows \begin{align} \alpha&=(z_0X-z_2,\;-(z_0\Theta-z_1),\;z_0^2\eps),\nonumber\\ \beta&=(z_0\Theta-z_1,\;z_0X-z_2,\;s),\qquad \gamma=-z_0^2\rho:M^{-1}\longrightarrow M^1. \label{phys:monad} \end{align} The matrix degree of \(\gamma\) is two and its coefficient degree is minus one. Hence it is part of the total differential, and \(d\gamma+\beta\alpha=0\). Projection to \(R\) gives a framing on \(2D\): near \(D=(z_0=0)\) the remaining two-variable Koszul complex is contractible, since one of \(z_1,z_2\) is invertible. For objects \(i,j\), the strict morphism model \(\widetilde{\mathcal C}_{ij}\) has generators \[ (g,r)\ \text{in degree }0,\qquad (a,b,u,v)\ \text{in degree }1,\qquad w\ \text{in degree }2. \] The source and target types are \(g,a,b,w:V_i\to V_j\), \(r:R_i\to R_j\), \(u:R_i\to V_j\), and \(v:V_i\to R_j\), with the normal twists in \eqref{phys:monad}. Its differential is \begin{align} D(g,r)&=(g\Theta_i-\Theta_jg,\ gX_i-X_jg,\ gs_i-s_jr,\ r\eps_i-\eps_jg) -g\rho_i+\rho_jg,\nonumber\\ D(a,b,u,v)&=aX_i-X_ja+\Theta_jb-b\Theta_i +u\eps_i+s_jv,\qquad Dw=0. \label{phys:hom-differential} \end{align} The last two terms of \(D(g,r)\) belong to the relation slot. Indeed, applying the second line to the four field entries gives \(g([\Theta_i,X_i]+s_i\eps_i) -([\Theta_j,X_j]+s_j\eps_j)g\), which is canceled by their coefficient differential. The map to monad morphisms sends \((g,r)\) to \(\operatorname{diag}(g,g,g,r,g)\), a field to minus the derivative of \eqref{phys:monad}, and \(w\) to the lower-to-top entry \(-z_0^2w\). Denote it by \(\kappa\). Its field entries, in the order lower, middle, frame, top, are \[ (-z_0b,z_0a,-z_0^2v),\qquad (-z_0a,-z_0b,-u). \] The only product of positive-degree generators that can be nonzero is \begin{equation}\label{phys:field-product} (a_2,b_2,u_2,v_2)(a_1,b_1,u_1,v_1) =w(-a_2b_1+b_2a_1-u_2v_1). \end{equation} Degree-zero products and their actions are ordinary rectangular composition. Equations \eqref{phys:hom-differential} and \eqref{phys:field-product} follow by multiplication of the displayed matrices. Thus \(\kappa\) is a morphism of differential graded categories. \begin{lemma}\label{phys:monad-hom} The morphism \(\kappa\) is a quasi-isomorphism on every universal chart and after arbitrary derived base change. The relative boundary-framing complex is its subcomplex \(r=0\). \end{lemma} \begin{proof} All entries in the matrix Hom complex have projective-plane twists between \(-2\) and \(2\), whose higher cohomology vanishes. At zero fields the monad is \(\OO\otimes R\oplus i_*V_0[-1]\). The Koszul resolution of the point \(i\) gives the four Hom blocks \[ \begin{array}{c|c} \text{source and target} & \text{nonzero cohomology}\\ \hline i_*V_i[-1],\,i_*V_j[-1]& \Hom(V_i,V_j)\otimes\bigwedge^kW\quad(k=0,1,2)\\ \OO R_i,\,i_*V_j[-1]&\Hom(R_i,V_j)\quad(k=1)\\ i_*V_i[-1],\,\OO R_j&\Hom(V_i,R_j)\Delta\quad(k=1)\\ \OO R_i,\,\OO R_j&\Hom(R_i,R_j)\quad(k=0). \end{array} \] For the first block, differentiation of the two Koszul equations gives the two exterior generators, and the lower-to-top map gives their exterior product. The two framing maps give respectively the restriction and the top Koszul class. Thus the displayed map \(\kappa\) identifies every generator with the corresponding class in this table. Combine simultaneous dilation in the two normal directions with the central \(V_0\)-gauge action. Each of \(\Theta,X,s,\eps\) has weight one, and \(\rho\) has weight two. The cone of \(\kappa\) is a perfect equivariant complex on the whole affine derived chart. If its classical support were nonempty, the positive dilation would put the origin in its closure. Its derived fiber at the origin is zero by the calculation above. Derived Nakayama then makes that support empty. This argument is made before imposing a stability open, and its quasi-isomorphism can subsequently be pulled back to any derived base. Boundary restriction is exactly \((g,r)\mapsto r\), proving the last assertion. \end{proof} Write \(\mathcal C_{ij}\) for \(r=0\). For arbitrary framing rank this means the fiber of the entire boundary restriction. For rank-one framing on the diagonal it is also the scalar trace fiber. \begin{lemma}\label{phys:proper-trace} The proper relative Serre trace on \(\mathcal C\) is the cochain pairing \begin{align} q_{\mathcal C}(g,w)&=q_{\mathcal C}(w,g)=\Tr(gw),\nonumber\\ q_{\mathcal C}((a,b,u,v),(a',b',u',v')) &=-\Tr(ab'-ba')-\Tr(v'u)+\Tr(vu'). \label{phys:raw-pairing} \end{align} The last two traces are taken on the respective framing spaces. After the shift \(T=\mathcal C[1]\), it agrees under \eqref{phys:dictionary} with the canonical critical pairing. \end{lemma} \begin{proof} Let \(\pi\) be the projection from the relative projective plane to its curve-and-bundle chart, and let \(\pi_D\) be its boundary restriction. Put \(G_j=R\pi_*\RHom(M,M(-jD))\). The second-order framing gives \(G_2\simeq G_1\), because its boundary cone is \(\operatorname{End}(R)\otimes R\pi_{D*}\OO_D(-D)=0\). The pairing in question is the specific product \[ G_1\otimes G_1\simeq G_2\otimes G_1 \longrightarrow G_3 \xrightarrow{\Tr,\ \mathrm{Serre}}\Delta[-2]. \] We compute its sign in the ordered cover \((z_0\ne0,z_1\ne0,z_2\ne0)\). Put \(x=z_1,y=z_2\), and order the five monad slots as lower, middle one, middle two, frame, top. At the origin the nonzero differential entries are \[ d_{21}=-y,\quad d_{31}=x,\quad d_{52}=-x,\quad d_{53}=-y. \] The contractions away from the point are \[ h_x=\frac{E_{13}-E_{25}}{x},\qquad h_y=-\frac{E_{12}+E_{35}}{y}, \] where \(E_{ij}\) is the elementary matrix. They satisfy \([d,h_x]=[d,h_y]=P_V\), the projector onto the unframed summand. Use the total differential \(\delta_{\mathrm{\check Cech}}+(-1)^p[d,-]\) in Čech degree \(p\). The two normal cocycles in \(G_1\) are \(T=E_{31}-E_{52}\) and \(X=-E_{21}-E_{53}\). A \(G_2\) lift of \(T\) has components \[ T/z_0,\qquad b_{01}=Th_x/z_0,\quad b_{02}=Th_y/z_0, \quad b_{12}=0,\qquad c_T=-h_x(b_{01}-b_{02}). \] Multiplication verifies \([d,b_{0i}]=-T/z_0\), \([d,c_T]+b_{01}-b_{02}=0\), and \[ \operatorname{str}(c_TX)=-\frac1{z_0xy}. \] For the section cocycle the corresponding components are \[ -E_{54}/z_0^2,\quad -E_{24}/(z_0^2x),\quad -E_{34}/(z_0^2y),\quad c_u=-E_{14}/(z_0^2xy). \] The dual section in \(G_1\) is \(v_1=-z_0E_{41}\), and \(\operatorname{str}(c_uv_1)=-1/(z_0xy)\). For the gauge class the top component of its lift is \(c_g=-h_xh_y/z_0^2\); multiplication by the relation \(-z_0E_{51}\) has supertrace \(+1/(z_0xy)\). Here the supertrace is \(-m_{11}+m_{22}+m_{33}+m_{44}-m_{55}\). Reversing the two field factors reverses their sign, whereas reversing gauge and relation does not. The ordered projective Serre trace sends \(1/(z_0xy)\) to \(+1\); this is the ordered residue convention of \cite[Proposition 5.2.3]{NayakSastry}. Tensoring these calculations with arbitrary rectangular Hom spaces gives exactly \eqref{phys:raw-pairing}. There is also a direct differential check. If \(P\) is minus the field pairing in \eqref{phys:raw-pairing}, cyclic trace gives \(P(Dg,z)=\Tr(gDz)\). For opposite gauge inputs \(g,h\), the additional ghost terms cancel as \[ \Tr\bigl((-g\rho_i+\rho_jg)h +g(-h\rho_j+\rho_ih)\bigr)=0. \] These identities verify the cochain relation with all ghosts. To see that the origin calculation determines the entire universal pairing, assign generator degree \(i\) the section-module weight \(-i\). The target \(\Delta[-2]\) has degree two and weight \(-2\). A pairing cochain of degree \(-k\), with \(r\) ghosts and ordinary polynomial degree \(n\), would satisfy \[ r=2+k-i-j,\qquad n+2r=2-i-j, \quad\text{hence}\quad n+r+k=0. \] For \(k>0\) there is no such cochain; for \(k=0\) only constants on complementary generator degrees occur. Finite semifreeness and exact invariants for the reductive frame group justify this calculation in derived Hom. Thus the comparison and its descent are fixed by the origin computation. The resulting universal maps pull back to every test algebra. Finally décalage, with \(s\) denoting suspension, gives \(\omega(su,sv)=(-1)^{|u|}q_{\mathcal C}(u,v)\). The tangent field form is consequently \(\Tr(d\Theta\wedge dX)+ds\wedge d\eps\), the derivative of \(-\Tr(X\,d\Theta)-\eps\,ds\). For a coordinate of degree \(a\), the canonical raw cross pairing of its cotangent generator with the base generator is \((-1)^{a+1}(-1)^a=-1\) times evaluation. The whole dual-half reversal in \eqref{phys:dictionary} therefore matches the gauge--relation block as well. \end{proof} \subsection{Closed forms and the determinant square} We specify the closed lift of the trace just calculated. In the convention of \cite[Theorem 2.12]{PTVV}, the leading component of \(\mathrm{Ch}_2\) is \(\tfrac12\Tr(\mathrm{mult})\). We therefore use \begin{equation}\label{phys:closed-normalization} \omega_{\Perf}^{\Tr}=2\mathrm{Ch}_2. \end{equation} On \(Y\), the given volume has \(K_Y=\OO_Y(-3D)\). The framed mapping construction of \cite[Theorem 3.3]{Spaide}, with extra divisor \(D\), applies: the lift of the specified framing from \(D\) to \(2D\) is étale because \(R\Gamma(D,\OO_D(-D))[1]=0\). Its framing isotropic path is zero, since the ideals of \(D\) and \(2D\) multiply to zero on \(3D\). The other path is the boundary triangle \(K_Y\to\OO_Y\to\OO_{3D}\). This constructs a closed \((-1)\)-shifted form whose leading pairing is precisely the proper product in Lemma~\ref{phys:proper-trace}. The factor in \eqref{phys:closed-normalization} does not change the coefficient \(1/2\) of \eqref{phys:potential}: the latter is its quadratic primitive coefficient. Let \(\operatorname{pr}:\mathcal C_{ij}\to\mathcal B_{ij}\) be the projection to the surface framing complex, forgetting \(b,v,w\). It is a strict functor. Its kernel has \(b,v\) in degree one and \(w\) in degree two, and has square zero by \eqref{phys:field-product}. The pairing \[ (b,v),(a,u)\longmapsto\Tr(ba)+\Tr(vu),\qquad w,g\longmapsto\Tr(wg) \] identifies this kernel with \(\mathcal B_{ji}^\vee K_C[-2]\). Writing \(\langle-,-\rangle\) for this pairing, the two bimodule identities are \[ \langle kh,b\rangle=\langle k,\operatorname{pr}(h)b\rangle,\qquad \langle hk,b\rangle=(-1)^{|h|(|k|+|b|)} \langle k,b\operatorname{pr}(h)\rangle. \] Here \(k\) lies in the kernel and \(h,b\) are homogeneous composable entries. In generator degree two they are rectangular trace identities, using \eqref{phys:field-product}; all other cases vanish. Thus, after curve sections and Serre duality, there is a natural paired triangle \begin{equation}\label{phys:paired-triangle} B_{ji}^{\vee}[-3]\longrightarrow C_{ij} \longrightarrow B_{ij},\qquad B_{ij}=R\Gamma(C,\mathcal B_{ij}),\quad C_{ij}=R\Gamma(C,\mathcal C_{ij}). \end{equation} It extends to finite twisted sums and cones by their block-matrix differentials. In particular it applies when a cofiber is a finite-length sheaf. The leading pairing comparison of Lemma~\ref{phys:proper-trace} extends to the full closed forms. Both forms have weight two for normal dilation. If \(\delta\) is their difference and \(h_0\) is its leading homotopy, then \(\delta-Dh_0\in F^3\DR\). For the Euler field \(\mathcal E\), Cartan's formula gives \begin{equation}\label{phys:cartan} H_0=h_0+\tfrac12\iota_{\mathcal E}(\delta-Dh_0) \in F^2\DR,\qquad DH_0=\delta. \end{equation} This uses the full de Rham mixed complex. In a formal presentation with invariant ideal \(J\), the filtration \(\prod_p J^{\max(m-p,0)}\Omega^p\) makes the operation continuous. Construct \eqref{phys:cartan} on the equivariant formal diagram, with its transition maps, before restricting to a completion at a moving point. For a diagonal object put \(B=B_{ii}\). The half line is \(\det B\); its square is given by \eqref{phys:paired-triangle}, duality, and the canonical critical factor \((-1)^{\operatorname{vdim}\cP_L}\) in \cite[Example 3.20]{KPS}. This defines the orientation used throughout the paper. The bimodule structure and the determinant double-sequence identity \cite[Lemma 2.4]{KPS} give its direct-sum and finite-flag compatibilities. \begin{proposition}\label{phys:orientation-square} Let \(\Psi\) be the equivalence of Proposition~\ref{phys:critical-equivalence}, or one of the supported equivalences constructed below, and let \(M_A,M_B\) be the orientation lines of its source and target. Its specified half-line comparison \(\gamma:M_A\to\Psi^*M_B\) satisfies \begin{equation}\label{phys:same-square} o_A=\det(d\Psi^*)\,o_B\,\gamma^{\otimes2}, \qquad d\Psi^*:\Psi^*\mathbb L_B\longrightarrow\mathbb L_A. \end{equation} Here \(o_A,o_B\) are the orientation square maps and \(\mathbb L_A,\mathbb L_B\) are cotangent complexes. \end{proposition} \begin{proof} First consider \eqref{phys:critical-equivalence}. Write its paired triangle as \(K\to C\to B\), with \(h:K\xrightarrow{\sim}B^\vee[-3]\). The cotangent triangle is \[ B^\vee[-1]\longrightarrow C^\vee[-1] \longrightarrow K^\vee[-1]. \] Identify its last object with \(B[2]\) by the inverse of \(h^\vee[-1]\), using the same shifted double-dual convention as in \cite[Appendix B.1]{KPS}. The physical and critical triangles have identical first arrows. On the last term, the dual of the equality \(h_{\mathrm{crit}}\Psi_K=h_{\mathrm{phys}}\), proved in Lemma~\ref{phys:proper-trace}, is \[ \Psi_K^\vee[-1]h_{\mathrm{crit}}^\vee[-1] =h_{\mathrm{phys}}^\vee[-1]. \] Consequently these are equivalent diagrams of fiber sequences. Taking their graded determinants, and inserting the same critical factor on both sides, proves \eqref{phys:same-square}. This keeps the direction of the cotangent derivative. For example, scaling \(h\) by \(c\) changes the raw square by \(c^{\chi(B)}\), and negating the whole dual half changes its derivative determinant by \((-1)^{\chi(B)}\); these are the same factor. Dualizing and shifting inverts the full determinant map. For a finite exact diagram, apply this proof to the paired bimodule triangles \eqref{phys:paired-triangle} on every entry of its double Hom grid. Naturality of duality and determinant double-sequence coherence give the asserted square for the resulting half-line map. The supported case and the identification of its full map with the derivative of the specified \(\Psi\) are spelled out in Lemma~\ref{phys:supported-determinant}. \end{proof} \subsection{The fixed positive quotient} We compare this orientation with the canonical one for \(N=K_C\oplus\OO_C\). For two fixed even-degree quotients \(q_i:N\to L_i\), put \(A_i=\ker q_i\), and assume their kernels are generically transverse. In the full deformation complex the dual-section and relation terms form a common isotropic subcomplex \(J\). Its orthogonal quotient is \[ J^\perp/J=\operatorname{End}(E)\otimes N[-1], \] and the two isotropic halves have quotients \(\operatorname{End}(E)\otimes A_i[-1]\). These are filtrations of paired complexes. \begin{lemma}\label{phys:fixed-quotient} The two orientation roots differ by \(\mathcal C_{\mathrm{res}}^{\,r^2}\), where \(\mathcal C_{\mathrm{res}}\) is one fixed quadratic line depending only on the two quotients. Choosing its quadratic trivialization once gives a square-preserving comparison, compatible with direct sums and flags. Its factors cancel canonically between a finite modification and its same-rank baseline. \end{lemma} \begin{proof} Refine the filtration by \(J\), the two isotropic halves, and \(J^\perp\). Determinant double-sequence coherence cancels the common hyperbolic part of their squares. The remaining ratio is the Serre linking form on the fixed finite module \(N/(A_0+A_1)\), of length \(m=\deg L_0+\deg L_1-2(g-1)\). Choose a local ordered frame with \[ A_1=\OO v_1,\quad A_0=\OO(uv_1+t^av_2),\quad v_1\wedge v_2=dt, \] where \(u\) is a unit. Decomposing \(xv_2\) along these two lines gives the linking form \[ b_{\mathrm{res}}(x,y)=\operatorname{Res} \frac{uxy\,dt}{t^a}. \] Adding an integral lift does not change the residue. In the basis \(1,t,\ldots,t^{a-1}\), its determinant is \((-1)^{a(a-1)/2}u(0)^a\). Tensoring with the trace form on \(\operatorname{End}(E)\) gives \[ \det b_E =(-1)^{mr(r-1)/2}(\det b_{\mathrm{res}})^{r^2}. \] The length \(m\) is even, so the additional sign is one. A change of frame acts by \(\operatorname{Ad}(g)\), whose determinant over the finite support ring is one; its norm is also one. Thus this is an equality of quadratic line maps over nonreduced bases, independent of \(E\) and of all its fields. The difference of the critical factors has exponent \(r^2(\deg L_1-\deg L_0)\), which is even. For a direct sum, opposite cross-Hom blocks give the factor \(\mathcal C_{\mathrm{res}}^{2r_1r_2}\), contracted by its quadratic form. The interchange sign is \((-1)^{mr_1r_2}=1\). The common determinant filtration gives coherence for further blocks and refinements. A modification and its baseline have the same rank, so their factors cancel by evaluation, including when the defect meets the residue divisor. \end{proof} \subsection{Support gluing and the derivative} For the fixed-point calculation use the equivariant projection \[ q_K:K_C\oplus\OO_C\longrightarrow K_C,\qquad S_K=\Tot_C K_C. \] Transport the fixed positive-quotient orientation by Lemma~\ref{phys:fixed-quotient}. Both quotient degrees are even, and their kernels are generically transverse: otherwise the component $\OO_C\to L$ of the positive quotient would vanish, forcing a surjection $K_C\to L$ contrary to $\deg L>\deg K_C$. The fixed quadratic factor has exponent \(r^2\) and cancels between a modification and its same-rank baseline. All Hom, section, support, and duality maps below are equivariant for the torus \(H\) defined next. Their determinant grids are taken weight by weight: weight zero supplies the orientation of the fixed enhancement, and the moving weights supply the localization index. Let \(Z_\lambda\subset\Tot_C(K_C\oplus\OO_C)\) be the monomial thickening specified by a partition \(\lambda\), and let \(I_0=\fib(\OO\to\OO_{Z_\lambda})\). Use the volume-preserving torus \[ H:(\theta,z)\longmapsto(a\theta,a^{-1}z). \] Consider a finite modification \(0\to\OO_{Z_\lambda}\to F\to\widetilde T\to0\). Ordinary \(H\)-fixed families of these modifications are identified by curve duality with quotients of the dualizing module \(\omega_{Z_\lambda}\), where \(\omega_{Z_\lambda}=\Hom_C(\OO_{Z_\lambda},K_C)\). Let two étale coordinate maps, over a smooth parameter scheme, identify selected neighborhoods of the finite defect support with neighborhoods on a reference curve. Their normal lifts have the form \[ (t,\theta,z)\longmapsto (f(t),\,\theta/f'(t),\,z), \] so they preserve \(dt\wedge d\theta\wedge dz\). The coordinate maps on ordinary quotient schemes will be denoted by \(\mathsf A,\mathsf B\). \begin{lemma}\label{phys:support-gluing} On the selected support branch there is a formal equivalence of the specified \(H\)-fixed physical enhancements whose truncation is \(\mathsf B\mathsf A^{-1}\) on every ordinary Artin base. It is constructed on all connective Artin bases by restriction and gluing with fixed determinant. \end{lemma} \begin{proof} For a rank-one perfect object define \[ \mathscr A_I=\fib(R\mathcal Hom_Y(I,I)\xrightarrow{\operatorname{str}}\OO_Y). \] The identity has supertrace one, so this is the strict trace kernel in a locally free model. Near \(\OO_D\), the perfect-object stack equals the derived Picard stack formally: derived Nakayama and Tor-amplitude zero make the deformation an invertible module. Moreover \(R\Gamma(Y,\OO_Y)\simeq R\Gamma(D,\OO_D)\). The determinant and boundary restriction square therefore identifies the boundary-framed formal problem with the fixed-determinant formal problem, including all higher automorphisms. If \(Z\) is the underlying finite defect support and \(U=Y-Z\), the modification problem is \begin{equation}\label{phys:modification} \bigl(\Perf(Y)\times_{\Pic(Y)}\{\det I_0\}\bigr) \mathop{\times}_{\Perf(U)\times_{\Pic(U)}\{\det I_0|_U\}} \{I_0|_U\}. \end{equation} It has deformation complex \(R\Gamma_Z\mathscr A_I\). An étale map on the selected finite support is universally injective and proper there, hence a closed immersion onto that support. Delete the other sheets over it. This gives an elementary distinguished square with the open complement. Descent for \(\Perf\) and \(\Pic\) gives an equivalence of \eqref{phys:modification} on the two sides. Its inverse is the Mayer--Vietoris homotopy fiber of the difference of restriction maps; the same fiber on mapping complexes gives full faithfulness and the higher cells \cite[Tag 08GS]{Stacks}. This construction uses the open complement in the derived ambient. For the monomial baseline, its transverse ideal has scalar endomorphisms, local \(\Ext^1\) equal to the Hilbert tangent representation, and no higher local Ext, by the two-variable Hilbert--Burch resolution. The two tangent characters at a box are \(a^{h_b}\) and \(a^{-h_b}\); none is zero-weight. After removing scalars, the \(H\)-invariant baseline complex on the complement and its overlaps is therefore acyclic. Curve sections and localization in curve points preserve these characters. For a square-zero extension of an Artin base, the obstructions, choices, and all automorphisms for forgetting the complement identification are shifts of this acyclic complex. Induction over small extensions and the Postnikov tower proves that forgetting it is a formal equivalence. It remains to identify the ordinary map. In an ordinary Artin family, the curve pushdown \(E\) is locally free. The monomial annihilator acts by zero away from the defect. A local curve parameter, or its monic characteristic polynomial, is a nonzerodivisor on the free curve module over an Artin ring. The annihilator consequently acts by zero everywhere. The section gives \(0\to\OO_{Z_\lambda}\to F\to\widetilde T\to0\) with base-flat finite cokernel. Relative curve duality gives its quotient of \(\omega_{Z_\lambda}\). Selected-sheet pullback commutes with these operations and with the section. Hence the truncation is exactly \(\mathsf B\mathsf A^{-1}\) on ordinary Artin families. \end{proof} \begin{lemma}\label{phys:supported-determinant} For the equivalence in Lemma~\ref{phys:support-gluing}, the algebraic half-line comparison defined by the four supported Hom and section maps has square \eqref{phys:same-square} for the derivative of that same equivalence. \end{lemma} \begin{proof} Write the surface modification triangle as \(G_0\to G\to T\). On the equivariant surface \(S_K\), put \[ B_G=\fib\bigl(\RHom_{S_K}(G,G)\to R\Gamma(S_K,G)\bigr), \quad M(G)=\det B_G,\quad M_H(G)=\det(B_G^H). \] Superscripts $H,+,-$ denote the invariant, positive-weight, and negative-weight parts for $H$. The map is precomposition by the section. For the paired triangle \eqref{phys:paired-triangle}, the positive-weight part of the physical cotangent complex has determinant \[ \det(\mathbb L_{\mathrm{phys}}^{>0}) =\det(B_G^-)\otimes\det(B_G^+). \] The localization orientation \(M(G)\otimes\det(\mathbb L_{\mathrm{phys}}^{>0})^{-1}\) therefore identifies with \(M_H(G)\), using the graded regrouping maps of \cite[Section 6.5]{KPS}. The double Hom grid is \(H\)-equivariant before taking determinants. Its full determinant identity is \begin{equation}\label{phys:four-hom} \frac{M(G)}{M(G_0)} = \det\RHom(T,G_0)\otimes\det\RHom(T,T) \otimes\det\RHom(G_0,T)\otimes(\det R\Gamma T)^{-1}. \end{equation} Its comparison maps are supported adjunction on \(\RHom(G_0,T)\), its Serre dual on \(\RHom(T,G_0)\), the support-equivalence unit and counit on \(\RHom(T,T)\), and composition of pushforwards on \(R\Gamma T\). They are algebraic and compatible with base change. Taking weight zero in each grid entry and then the ordered determinant product defines \(\gamma\) on the lines \(M_H\). The physical triangle is \(I_0\to I\to J\) with \(J=\widetilde T[-1]\), since the cofiber of the map of source sections is zero. Represent \(I\) as the twisted sum with differential \[ D_I=\begin{pmatrix}d_0&\kappa_0\\0&d_J\end{pmatrix}. \] For a degree-\(k\) endomorphism \(M=\begin{pmatrix}a&b\\c&e\end{pmatrix}\), its differential has blocks \begin{align*} (\delta M)_{00}&=d_0a-(-1)^kad_0+\kappa_0c,\\ (\delta M)_{0J}&=d_0b-(-1)^kbd_J +\kappa_0e-(-1)^ka\kappa_0,\\ (\delta M)_{J0}&=d_Jc-(-1)^kcd_0,\\ (\delta M)_{JJ}&=d_Je-(-1)^ked_J-(-1)^kc\kappa_0. \end{align*} Graded cyclicity cancels the extension terms in the supertrace. The strict trace kernel has a three-step filtration by the upper-right block, the trace-zero upper triangular blocks, and the whole kernel. Its graded pieces are \begin{equation}\label{phys:trace-filtration} R\mathcal Hom_Y(J,I_0),\qquad \mathscr A_{I_0}\oplus R\mathcal Hom_Y(J,J),\qquad R\mathcal Hom_Y(I_0,J). \end{equation} The middle identification sends \((a_0,e)\) to \((a_0-\operatorname{str}_J(e)\id_{I_0},e)\). The extension differential is retained in the filtered complex. After supported sections and \(H\)-invariants, the baseline term in \eqref{phys:trace-filtration} is acyclic and the other terms are perfect with finite support, so their determinants are defined. Restriction and gluing preserve the displayed filtration. Consequently its determinant map is the determinant of the derivative of the functor in Lemma~\ref{phys:support-gluing}. To compare with the proper Hom grid, use the natural maps from supported to proper sections on its surviving finite-support terms. The possible trace-projector correction on \(\RHom(J,J)\) is \(1-uv\), where \(u(1)=\id_J\) and \(v=\operatorname{str}_J\). Since \(vu=\rk(J)=0\), the identity \(\det(1-uv)=\det(1-vu)\) gives determinant one, even on nonreduced bases. The paired triangle \eqref{phys:paired-triangle}, applied to the same exact arrow diagram, has surface halves \[ \RHom(T,G_0),\quad \RHom(T,T),\quad \fib(\RHom(G_0,T)\to R\Gamma T). \] Its full half determinant is \eqref{phys:four-hom}; its weight-zero half determinant is the same formula with invariants in every Hom and section term. Applying the proof of Proposition~\ref{phys:orientation-square} to this grid identifies the square of \(\gamma\) with the derivative determinant just computed. The three full surface Hom Euler pairings are zero, and the section term has Euler characteristic \(d=\operatorname{length}T\). If \(m_0,m_+,m_-\) are the Euler changes of the half complex in weight zero, positive weights, and negative weights, respectively, then \[ m_0+m_++m_-=-d. \] The fixed determinant square retains the dual-half reversal \((-1)^{m_0}\). The moving weight pieces retain their corresponding reversal factors; their product with the fixed factor is \((-1)^d\). These factors are independent of the even residue length in Lemma~\ref{phys:fixed-quotient}. \end{proof} \subsection{The comparison over the full parameter ring} The preceding construction also compares the closed forms. On full endomorphism complexes, stable Koszul local cohomology gives the supported product. The boundary section is invertible on the support. Denote the maps from supported sections to \(R\Gamma\RHom(I,I(-jD))\) by \(J_j\). They satisfy \(J_i(a)J_j(b)=J_{i+j}(ab)\). Restricting the inputs to the trace kernel only after taking these products gives the proper pairing of Lemma~\ref{phys:proper-trace}. The selected étale maps preserve the volume. Their canonical bundle maps and duality counits therefore compare this pairing on the full moving family, including directions that move the support. The residue change-of-generators, localization, and base-change identities in \cite[\S\S4,10]{NayakSastry} apply to these counits. Locally a monic characteristic polynomial in the curve coordinate, together with high powers of the two normal coordinates, gives a bounded flat Koszul model. Lifting its coefficients to a connective presentation gives the same construction after derived base change. Different lifts compute the same support functor and its already defined duality counit. Normal dilation gives both closed forms weight two. Equation \eqref{phys:cartan} consequently constructs their full closed homotopy on the common parameter diagram. The following argument upgrades this statement to the particular ordinary maps used in coefficient descent. \begin{proposition}\label{phys:parameter-square} On a neighborhood of the punctual locus, the maps \(\mathsf A,\mathsf B\) identify the algebraic \(d\)-critical sections and the oriented vanishing-cycle coefficients. The comparison holds over the entire smooth moving-support and coordinate parameter space. \end{proposition} \begin{proof} The ordinary map \(\mathsf A\) is étale. Its unique derived étale lift relative to the specified physical enhancement is identified with the lift in Lemma~\ref{phys:support-gluing}: their ordinary maps agree and their lifting spaces agree on all Artin bases. On a selected completion this makes the \(\mathsf A\)-side the fixed physical formal problem extended over \(R=\C[[s_1,\ldots,s_e]]\). Choose one minimal critical model at its reference point and extend scalars: \begin{equation}\label{phys:minimal-model} \mathcal A_R=R[[x_1,\ldots,x_m]][e_1,\ldots,e_m], \quad |e_j|=-1,\quad de_j=I_j(x)\in(x)^2. \end{equation} Put \(I=(I_1,\ldots,I_m)\subset R[[x_1,\ldots,x_m]]\). The existence of such a model is the formal critical-chart statement of \cite{BBJ}; it is chosen once over \(\C\). Represent the two specified maps on cofibrant complete models, calling the representatives \(\alpha,\beta\). Let \(f_A,f_B\) be their chart potentials and \(\lambda_A,\lambda_B\) the Liouville one-forms of their critical models. Set \[ F=\alpha^*f_A-\beta^*f_B,\qquad \ell_0=\alpha^*\lambda_A-\beta^*\lambda_B. \] The preceding closed homotopy makes \(F-\ell_0+H_0\) closed in the full relative de Rham complex. Its augmentation to \(R\) is zero: both potentials are normalized to vanish on their reduced critical loci, and the moving section is a critical family over the reduced ring \(R\). Give \(x,e,d_{\DR}x,d_{\DR}e\) Euler degree one and \(R\) degree zero. On positive Euler degree \(v\), \(\iota_{\mathcal E}/v\) contracts the augmented de Rham differential. The internal differential raises that degree, by \eqref{phys:minimal-model}. The perturbation series \[ h_{\mathcal E}(1+dh_{\mathcal E})^{-1} \] therefore converges in the combined adic and Hodge filtration and contracts the full augmented complex. The zero- and one-form components of a primitive satisfy \[ F=d\Psi_0,\qquad -\ell_0=d_{\DR}\Psi_0+d\psi_1. \] A degree-zero chart coordinate pulls back to a function of \(x\) over \(R\); hence \(\ell_0\) has no \(d_{\DR}e_j\) component. Degree constraints give \[ \Psi_0=\sum_jL_je_j,\qquad \psi_1=\sum_{i,j}M_{ij}e_i\,d_{\DR}e_j +\sum_{i,j,k}N_{ijk}e_ie_j\,d_{\DR}x_k. \] Comparing \(d_{\DR}e_j\) coefficients yields \(L_j+\sum_iM_{ij}I_i=0\). Consequently \begin{equation}\label{phys:ideal-square} F=-\sum_{i,j}M_{ij}I_iI_j\in I^2 \quad\text{in }R[[x]]. \end{equation} This is a single identity over the full parameter ring. For original charts with acyclic Hessian variables \(y\), put \(J=(I,y)\). A class modulo \(J^2\) has the form \(f_0(x)+\sum y_af_a(x)\). If its differential vanishes modulo \(J\), its \(dy_a\) coefficients give \(f_a\in I\). Thus \eqref{phys:ideal-square} detects equality in the original charts, with their Hessian factors retained. The product parameter directions contribute the identical factor \(K_{\mathrm{par}}\) to both half lines. The maps are over the parameter base, so their derivative matrices have an identity parameter block and introduce no additional determinant. The two \(d\)-critical sections and the half-line map of Lemma~\ref{phys:supported-determinant} existed algebraically before completion. Faithful flatness of local completion detects equality of their sections of \(\OO/J^2\), and then equality of the two existing square maps on the reduced base. The identification of the critical cotangent determinant with the \(d\)-critical canonical line is \cite[\S\S3.5--3.6]{KPS}, with the same Hessian and critical factor used above. Equation \eqref{phys:same-square} is therefore the oriented \(d\)-critical comparison, proving the assertion about vanishing-cycle coefficients. \end{proof} \section{Monodromic coefficients and localization}\label{phys:mmhm-section} The intermediate comparisons take place in monodromic mixed Hodge modules. We give the descent details needed for the stack correspondences. For an oriented critical chart \((U,f)\) the normalized coefficient is \[ \Phi_{(U,f)} =\phi_f^{\mathrm{mon}}(\Q_U)\La^{-\dim U}, \] tensored with its orientation torsor. Vanishing cycles are normalized to be perverse exact and to satisfy \(\phi_0=\id\). All products in this formula are monodromic products. The ordinary Hodge category embeds exactly and symmetrically monoidally in this category \cite[\S2.1]{DM}. \subsection{An explicit coefficient projector} Let \(\pi:V\times\A^1\to V\), and let \(B_V\) be the stable geometrically monodromic Hodge category on \(V\times\A^1\). The constants form the image of \(\pi^*\). Since \(\id\to R\pi_*\pi^*\) is an equivalence, the functor \begin{equation}\label{phys:projector} J_VF=\cofib(\pi^*R\pi_*F\to F) \end{equation} is the reflector onto \(\ker R\pi_*\). Indeed, \(R\pi_*J_VF=0\), and for \(R\pi_*K=0\) adjunction gives \(\operatorname{Map}(J_VF,K)\simeq \operatorname{Map}(F,K)\). Thus the quotient by constants is this explicit kernel. This projector also identifies the heart. If \(j:\Gm\into\A^1\), \(i:\{0\}\into\A^1\), the triangle \[ \Q_{\A^1}\longrightarrow i_*\Q \longrightarrow j_!\Q_{\Gm}[1] \] and the change of coordinates \((u,v)\mapsto(u+v,u)\) give \begin{equation}\label{phys:DM-projector} J_VF\simeq R(\id_V\times +)_*(F\boxtimes j_!\Q_{\Gm}[1]). \end{equation} The first map of the resulting triangle is exactly the counit in \eqref{phys:projector}. Equation \eqref{phys:DM-projector} is the perverse-exact projector of \cite[\S2.1, Lemmas 2.1--2.6]{DM}. Its image has the usual monodromic perverse heart. Both formulas commute with smooth pullback, and membership in the kernel is smooth local. They therefore descend through the ordinary smooth nerve of an Artin stack, using the Hodge six operations of \cite{Tubach}. The nilpotent and semisimple monodromy operators remain in this construction, as do the Hodge and weight filtrations. \subsection{The integral localization map} For a derived stack $X$, define the filtered and graded stacks by \begin{equation}\label{phys:filtered-stacks} \operatorname{Filt}(X)=\operatorname{Map}([\A^1/\Gm],X), \qquad \operatorname{Grad}(X)=\operatorname{Map}(B\Gm,X). \end{equation} Restriction at zero and evaluation at one give $\operatorname{Grad}(X)\xleftarrow{\operatorname{gr}} \operatorname{Filt}(X)\xrightarrow{\operatorname{ev}}X$. We use the component specified by the chosen integral grading. For a one-parameter torus action $\mu$, write $X^\mu$ for the fixed locus and $X^{\mu,+},X^{\mu,-}$ for the attracting loci as the parameter tends to zero and infinity. Their limit and evaluation maps are \[ X^\mu\xleftarrow{p^\pm}X^{\mu,\pm} \xrightarrow{\eta^\pm}X. \] Distinguish the functors \[ \operatorname{Loc}_\mu =(p^+)_*(\eta^+)^!\simeq(p^-)_!(\eta^-)^*, \qquad A_\mu=(p^+)_!(\eta^+)^* =\operatorname{Loc}_{\mu^{-1}}. \] The first is used for filtered-stack attachment, and the second for compact attracting strata. At a fixed point, the index convention is \[ I_\mu= \bigl(\rk^+H^0(T_X)-\rk^-H^0(T_X)\bigr) -\bigl(\rk^+H^{-1}(T_X)-\rk^-H^{-1}(T_X)\bigr), \] as in \cite[Definition 7.21]{KPS}. For the surface-half model \eqref{phys:paired-triangle}, it is \(\chi(B^-)-\chi(B^+)\). \begin{proposition}\label{phys:integral-localization} For the oriented \((-1)\)-shifted symplectic stacks used here, the integral map is an isomorphism in monodromic mixed Hodge modules \begin{equation}\label{phys:integral-map} \zeta_X: \Phi_{\operatorname{Grad}(X)}\La^{-I_\mu} \xrightarrow{\ \sim\ } R\operatorname{gr}_*\operatorname{ev}^!\Phi_X. \end{equation} It is compatible with products, smooth pullback, and matching refinements of an ordered finite flag. For a scheme with a torus action, its compact attracting version is \begin{equation}\label{phys:compact-attractor} (p^+)_!(\eta^+)^*\Phi_X \simeq\Phi_{X^\mu}\La^{I_\mu}. \end{equation} \end{proposition} \begin{proof} The stacks in use are covered by finite-type quotient charts, are quasi-separated, and have affine stabilizers. Their orientations and their product and flag maps are supplied by \eqref{phys:paired-triangle} and Proposition~\ref{phys:orientation-square}. These are the stack hypotheses of \cite[Corollary 7.19]{KPS}. We describe the mixed-Hodge construction of its map; this also fixes the normalization. On an equivariant critical chart, use the specialization exchange constructed by units and counits in \cite[Theorem 3.7, Proposition 6.2]{Descombes}. Its staircase inverse follows from the adjunction triangle identities. Tensor it with the finite-torsor map of \cite[(6.75)]{KPS}. The normalization on a moving hyperbolic quadratic block is the compact Thom map of the same repelling axis as in \cite[Theorem 7.9]{KPS}. Switching that axis would introduce \((-1)^r\) in rank \(r\). The scalar is checked on the normalized quadratic point, where the constructed maps lie in the perverse heart. All other maps are transformations in the Hodge six-functor system. For a self-dual filtration, locally choose a splitting and write \(q=q_{\mathrm{gr}}(1+N)\), with \(N\) filtration-raising and self-adjoint. The finite polynomial \((1+N)^{-1/2}\) orthogonalizes it. Orthogonal splittings form a torsor under the filtration-raising orthogonal unipotent group, an iterated affine-bundle torsor. Pullback to it is fully faithful, since its direct-image adjunction unit is an equivalence. The constructed map and all its comparison squares descend uniquely. The interpolation \(\exp(t\log u)\) between splittings has determinant one, so it contributes no orientation character. For smooth descent, let \(q:U\to X\) be an equivariant smooth chart and let \(q_\mu:U^\mu\to X^\mu\) be its fixed-locus map. Let the relative tangent of \(q\) have weight-zero, positive-weight, and negative-weight ranks \(d_0,d_+,d_-\), and let \(d=d_0+d_++d_-\). For a smooth map \(h\) of relative dimension \(e\), purity is \(h^!=h^*[2e](e)\). Applied to the fixed and attracting maps, it gives \[ q_\mu^*\operatorname{Loc}_X \simeq\operatorname{Loc}_\mu q^*[2d_-](d_-) =\operatorname{Loc}_\mu q^*\La^{-2d_-}. \] With normalized smooth pullback \(q^\dagger=q^*\La^{-d}\), this becomes \[ q_\mu^\dagger\operatorname{Loc}_X \simeq\operatorname{Loc}_\mu q^\dagger \La^{d_+-d_-}. \] The indices obey \(I_U=I_X+d_+-d_-\), so both the shift and Tate factor of the local map descend. The common charts and their fixed overlaps are \cite[Proposition 5.27, Corollary 5.28]{KPS}. On every level \(p_m\) of their smooth nerve, the two ordinary pullbacks are the same invertible shift and twist of perverse heart objects. Hence their mapping spaces have \(\pi_r=\Hom(A,B[-r])=0\) for \(r>0\). The smooth comparison squares, which are purity and Beck--Chevalley squares, give equal maps on double overlaps. Their compositions give the higher equalities. The discrete mapping spaces then descend the isomorphism and its inverse through the full nerve. Degeneracy compatibility is obtained by pullback of these identities, without applying smooth purity to a nonsmooth diagonal. For two ordered cuts, write their block weights as \(a_i,b_i\) in the same order. Then \((a_i-a_j)(b_i-b_j)\ge0\) on every cross-Hom block. This checks matching on the full cotangent complex, including relation and obstruction terms. Finitely many ordered cuts admit a common multi-torus chart by the construction of \cite[Lemma 5.30]{KPS}. On this locus the multi-filtered square is Cartesian and its excess map is the identity. Chart comparisons reduce to ordinary Beck--Chevalley by \cite[Lemma 7.4]{KPS}. Thus the product and matching-refinement constructions in \cite[Propositions 7.10--7.12, Lemmas 7.14--7.15]{KPS} are the same Hodge transformations, with determinant associators from \eqref{phys:paired-triangle}. This proves \eqref{phys:integral-map}. Using the inverse action changes the index sign and gives \eqref{phys:compact-attractor}. \end{proof} \begin{lemma}\label{phys:closed-counit} Suppose a filtered correspondence \(G\xleftarrow{q}F\xrightarrow{p}U\) occurring in \eqref{phys:integral-map} has classical evaluation a closed immersion \(i:F_{\mathrm{cl}}\into U_{\mathrm{cl}}\). Let \(h_U:U\to B\) and \(h_G:G\to B\) satisfy \(h_Up=h_Gq\). Its induction map, formed from the adjoint of \(\zeta\), is, after that isomorphism, \[ Rh_{U*}(i_*i^!\Phi_U\longrightarrow\Phi_U). \] Its cofiber is \(Rh_{U*}Rj_*j^*\Phi_U\), where \(j\) is the complementary open immersion. \end{lemma} \begin{proof} Put \(K_G=\Phi_G\La^{-I}\), and let \(c:q^*K_G\to p^!\Phi_U\) be the adjoint of \(\zeta:K_G\to Rq_*p^!\Phi_U\). The triangle identity is \(Rq_*c\circ\eta_q=\zeta\). Consequently induction is the composite \[ Rh_{G*}K_G\xrightarrow{Rh_{G*}\zeta} Rh_{U*}Rp_*p^!\Phi_U \xrightarrow{Rh_{U*}\epsilon_p}Rh_{U*}\Phi_U. \] Here \(p_*=p_!\) because its classical map is the stated closed immersion. The last arrow is therefore its actual counit. The closed/open localization triangle proves the cofiber assertion. No properness of \(h_U\) is needed. \end{proof} \subsection{One-sided constructibility on finite-type windows} We will apply the preceding operations on bounded finite-type stack windows. If \(V\to\mathcal X\) is a finite smooth atlas, \(p_m:V_m\to\mathcal X\) its ordinary nerve, and \(f_m:V_m\to B\) the maps to a finite-dimensional base, then a common lower standard bound \(b\) for \(p_0^*K\) is a lower bound for every \(p_m^*K\). Ordinary pullback is standard \(t\)-exact and \(Rf_{m*}\) preserves this lower bound. For \(C^m=Rf_{m*}p_m^*K\), the normalized term \(N^mC\) is a retract of \(C^m\), and the totalization tower has fiber \[ \fib(\operatorname{Tot}_m C\to\operatorname{Tot}_{m-1}C) \simeq (N^mC)[-m]\in D^{\ge b+m}. \] Thus each fixed degree depends on a finite partial totalization. Those finite terms have constructible cohomology. The finite amplitude difference between the standard and perverse structures on \(B\) gives finite constructible perverse cohomology and a one-sided bound. In a specified output range, finite convolutions can likewise be computed after a finite Postnikov truncation. \section{Harder--Narasimhan restrictions} \label{sec:comparison} We first construct the restriction maps that will be inverted in a perverse quotient. All constructions in this section are made separately on $X_0$ and $X_1$. The identification of either physical moduli problem with its critical presentation, including its orientation, is the one constructed in Section~\ref{sec:physical}. \subsection{The framed heart and its numerical bounds} Let $X=\Tot_C(N)$ be either threefold in \eqref{eq:physical-rows}, and let $Y=\mathbf P_C(\OO_C\oplus N)$ with infinity divisor $D_\infty=\mathbf P_C(N)$. In the usual tilted heart on $Y$ consider \begin{equation}\label{cmp:heart} \mathscr A_X=\langle\OO_Y, \operatorname{Coh}_{c,\leq1}(X)[-1]\rangle_{\mathrm{ex}}. \end{equation} The framing is an isomorphism along $D_\infty$, retained over the test base. In framing rank one, the weak slope of a framed object is a real parameter $t$. For every nonzero unframed coherent sheaf $F$, define $\mu(F[-1])=\chi(F)/\rk_C(F)$ when $\rk_C(F)>0$, and define $\mu(F[-1])=+\infty$ when $\rk_C(F)=0$. Weak semistability means $\mu(J)\leq\mu(Q)$ for every exact sequence $0\to J\to I\to Q\to0$. The heart criterion applies because $\operatorname{End}(\OO_Y)=\C$ and \[ \operatorname{Hom}(\OO_Y,F[-1])=0, \qquad \operatorname{Hom}(F[-1],\OO_Y) =\Ext^1_Y(F,\OO_Y)=0. \] The latter vanishing follows from codimension at least two and the local-to-global Ext spectral sequence. Thus the heart criterion of \cite[Proposition 3.6]{TodaPT} applies. To check noetherianity, consider a chain of epimorphisms. Its nonnegative framing and curve ranks eventually stabilize, so its remaining kernels are zero-dimensional objects shifted by $[-1]$. The ordinary $H^1$ sheaves form an eventually constant chain of quotients, after which the $H^0$ sheaves form an ascending chain inside the fixed reflexive hull of the first one. Coherence terminates this chain. The corresponding decreasing-chain condition for weak Harder--Narasimhan filtrations follows because a remaining nonzero zero-dimensional quotient has slope $+\infty$. These are the two chain conditions in \cite[Proposition 2.12]{TodaPT}. The boundary framing also removes infinitesimal Picard directions. For the projection $\pi:Y\to C$, relative projective-space cohomology gives \begin{equation}\label{cmp:boundary-trace} R\pi_*\OO_Y(-D_\infty)=0. \end{equation} This equality is preserved by derived base change. Hence the trace deformation complex preserving the specified boundary trivialization vanishes. Write $\mathscr X_{r,\chi}(c)$ for the physical window whose projected $L$-Higgs bundle is pure with minimum Euler slope greater than $c$. Its nonzero-section open is denoted by $\mathscr X_{r,\chi}^{\times}(c)$. \begin{lemma}\label{cmp:cyclic-bound} A compact cyclic structure sheaf $\OO_Z$ of curve rank $e$ satisfies \[ \chi(\OO_Z)\geq a_e. \] Every final $L$-Higgs Harder--Narasimhan quotient of its pushdown is again the pushdown of a cyclic physical quotient. Consequently its minimum Euler slope is at least $\alpha_e$. A positive-chamber semistable framed factor of curve rank $e$ has pure projected bundle with minimum Euler slope at least $\alpha_e$. \end{lemma} \begin{proof} Write $\Hom_L$ for morphisms of $L$-Higgs bundles. The negative line $A=K_CL^{-1}$ preserves the projected Higgs filtration. At a strictly decreasing cut $E'\subset E$, the off-diagonal transverse field lies in $\Hom_L(E',(E/E')\otimes A)=0$. In the nonsplit row, two local transverse lifts differ by a multiple of the projected field $\Theta$; their off-diagonal blocks therefore agree. The vanishing is global. The same argument applies to an equal-slope subobject of a semistable factor. Filter the cyclic module of $\OO_Z$ by monomials of total degree at most $j$ in the two normal variables and saturate this filtration on $C$. Let its positive successive ranks be $r_0,\ldots,r_s$. They occur consecutively: a stage at which the cyclic span stops growing is already preserved by both variables. Thus $s\leq e-1$, $\sum r_j=e$, and \[ \sum_{j=0}^{s}jr_j\leq\frac{e(e-1)}2. \] Every line summand of $\Sym^j(N^\vee)$ has degree at least $-j\ell$ in either row. A rank-$r_j$ quotient has degree at least $-j\ell r_j$, and saturation increases degree. Therefore \[ \deg\pi_*\OO_Z\geq-\frac{\ell e(e-1)}2, \qquad \chi(\OO_Z)\geq-he-\frac{\ell e(e-1)}2=a_e. \] By the negative-line argument, a final Higgs quotient is a physical quotient. A quotient of a cyclic structure sheaf is another cyclic structure sheaf. Applying the preceding bound to its rank $f\leq e$ gives slope at least $\alpha_f\geq\alpha_e$. The coherent projection of an object with $H^0(I)=I_Z$ and $H^1(I)=Q$ is the exact sequence \begin{equation}\label{cmp:projection-sequence} 0\longrightarrow\pi_*\OO_Z\longrightarrow E_I \longrightarrow\pi_*Q\longrightarrow0. \end{equation} It follows by applying $R\pi_*(-D_\infty)[1]$ to the truncation triangle and using \eqref{cmp:boundary-trace}. Finite-$t$ semistability forces $E_I$ to be pure. Indeed its maximal zero-dimensional submodule lifts uniquely through the coherent framing row because $\Hom_S(TK_S^{-1},\OO_S)=\Ext^1_S(TK_S^{-1},\OO_S)=0$ on the smooth surface $S=\Tot_C(L)$. The lift is preserved by the field, and would give an unframed subobject of infinite slope. Every positive-rank quotient factor of $Q$ has slope at least $t$. Equation~\eqref{cmp:projection-sequence} now gives the lower bound $\min(\alpha_e,t)=\alpha_e$ when $t>0$. \end{proof} Purity and a lower minimum-slope bound are extension closed. Thus every extension of the good graded factors in the lemma lies in the prescribed window. This applies to the entire attracting fiber. The same assertion holds for the immediate negative chamber $t=-\epsilon$, $0<\epsilon<1/r$: a finite slope with denominator at most $r$ that is at least $-\epsilon$ is nonnegative. \subsection{Complete coherent flags} The diagonal resolution of Section~\ref{sec:physical} realizes framing rank at most one by coherent diagrams on $S$ with row \begin{equation}\label{cmp:coherent-row} 0\longrightarrow\OO_S\longrightarrow U \longrightarrow FK_S^{-1}\longrightarrow0, \end{equation} where $F$ denotes the surface projection of the physical sheaf. The diagrams also include the compatible global field arrows. The nonsplit torsor changes the extra arrow but not this exact row. Exact sequences and flat families are termwise exact; in the split case this is also the construction of \cite[Theorem 4.5.7 and Corollary 4.5.9]{TodaCat}. \begin{lemma}\label{cmp:complete-receiver} For fixed graded Hilbert polynomials and a fixed position of the framing, compatible complete coherent flags in \eqref{cmp:coherent-row} have a representable projective evaluation map to the middle object. The receiver includes nonsaturated subobjects and torsion in the graded quotients. \end{lemma} \begin{proof} Use the fixed compactification $\overline S=\mathbf P_C(\OO_C\oplus L)$. The compactly supported quotient $F$ extends by zero, as do its field targets, and the rank-one part of $U$ extends as $\OO_{\overline S}$. Form the product of the relative projective flag-Quot schemes of these coherent terms. The flag of $\OO$ is zero before the framing position and is all of $\OO$ afterwards. Require preservation by every field arrow and containment of this framing by setting the composites to the universal quotient sheaves equal to zero. These are closed incidence conditions. For example, a common sufficiently positive twist on the bounded flag-Quot family makes them zero entries of matrices between base-change-compatible vector bundles. The incidence conditions imply row exactness. Before the framed step, the flag term of $U$ has support of dimension at most one, so it intersects the torsion-free $\OO_{\overline S}$ trivially. Its map to the corresponding term of $FK_S^{-1}$ is injective; the prescribed Hilbert polynomials make it an isomorphism on every fiber. After the framed step, divide by the contained $\OO$ and apply the same argument. Flatness of the flag terms and quotients upgrades these fiberwise identifications to exact rows after arbitrary base change. All changes from $\OO$ are supported on $\supp F$. Thus the boundary flag is the fixed zero or $\OO$ flag. The resulting closed incidence subscheme is projective over the middle object. Its field arrows are induced by those of the middle diagram, including on a nonsaturated inclusion. \end{proof} For the rank-zero framed semistable factor at finite $t$, set $a_0=0$. This factor is the vacuum $\OO_Y$, with Euler characteristic $\chi_0=0$. \begin{lemma}\label{cmp:finite-types} For fixed $r,\chi,c$, a finite list of coherent receivers suffices for the Harder--Narasimhan decomposition in a sufficiently large positive chamber. A single chamber parameter can be chosen after the complete receivers, including their torsion quotients, have been formed. \end{lemma} \begin{proof} Choose a rational $\alpha\leq\min(c,\alpha_r,0)$. If $E'$ is a pure rank-$f$ subobject of the projected middle bundle, its torsion-free quotient has Euler characteristic at least $\alpha(r-f)$. Hence \begin{equation}\label{cmp:prefix-bound} \mu(E')\leq\frac{\chi-\alpha(r-f)}{f},\qquad b=\max_{1\leq f\leq r}\frac{\chi-\alpha(r-f)}f. \end{equation} For $t>\max(b,0)$ the first factor is framed; an earlier unframed factor would violate \eqref{cmp:prefix-bound}. Write its class as $(1,e,\chi_0)$ and the successive unframed classes as $(0,r_j,\eta_j)$. They satisfy \begin{equation}\label{cmp:finite-list} \begin{gathered} a_e\leq\chi_0\leq\chi-\alpha(r-e),\qquad \eta_j>cr_j,\qquad \sum_jr_j=r-e,\qquad \chi_0+\sum_j\eta_j=\chi. \end{gathered} \end{equation} This is a finite list. The strict inequalities are retained in the list; $\alpha$ is only a bounding device. The projected window is bounded. Its Higgs Harder--Narasimhan numerical list is finite, and successive slopes in the ordinary bundle filtration of a semistable $L$-Higgs factor have gaps at most $\ell$. In particular a rank-$f$ semistable Higgs factor of Euler slope $\mu$ has underlying minimum degree slope at least $\mu+h-(f-1)\ell$. The coherent extensions and compatible arrows in \eqref{cmp:coherent-row} form finite-type families, being represented by bounded perfect relative Hom complexes. Form the projective receivers of Lemma~\ref{cmp:complete-receiver} for the finite list \eqref{cmp:finite-list}. Their graded coherent sheaves form bounded families. For a common large twist $k$, the length of a maximal zero-dimensional subsheaf $T_0(G)$ is at most $h^0(G(k))$, uniformly bounded. For each possible length, its possible positions are covered by the relative Quot scheme with the complementary Hilbert polynomial. This avoids requiring the maximal torsion subsheaf itself to be flat in the family. Absorb torsion backwards in each flag; each packet moves left through at most $r$ positive-rank steps. Repeating the finite Quot construction still gives bounded families. If $H$ is a pure Higgs subobject of $G/T_0(G)$, then \[ \chi(H(k))\leq h^0(H(k)) \leq h^0((G/T_0(G))(k))\leq h^0(G(k)), \] since $H^1(T_0(G))=0$. Thus all slopes occurring after torsion absorption and Higgs refinement have a common upper bound $b_{\mathrm{fl}}$. Choose a generic $T$ above $b$, $b_{\mathrm{fl}}$, and the final stable-pair thresholds for the finitely many framed classes. These thresholds can be obtained directly: if $t>\chi-a_r$, a positive-rank part of $H^1(I)$ would have Euler characteristic at least $t$, contradicting \eqref{cmp:projection-sequence} and Lemma~\ref{cmp:cyclic-bound}. The remaining $H^1(I)$ is zero-dimensional, and $\Ext^2(Q,\OO_X)=0$ on the smooth threefold lifts the truncation triangle to an ordinary pair. Absence of zero-dimensional subobjects makes its sheaf pure. Conversely the pure subobject slopes of a bounded stable-pair family are uniformly bounded. Thus $T$ is a stable-pair chamber simultaneously for every receiver under consideration. \end{proof} \subsection{The closed-stratum counit} Order the finite types first by increasing framed curve rank $e$, then by decreasing anchored tail polygon, choosing a linear extension of this order. Its vertices are \[ (e,\chi_0),\quad (e+r_1,\chi_0+\eta_1),\quad\ldots,\quad(r,\chi). \] The initial height $\chi_0$ is part of the polygon. Absorbing torsion of length $d$ into the preceding flag member raises the preceding vertex by $d$ and leaves later vertices fixed; this includes the initial vertex. Higgs refinement replaces the result by its concave majorant. If the framed prefix is unstable, its first framed factor has smaller curve rank and is earlier in the primary order. The uniform choice of $T$ ensures that every unframed tail slope is less than $T$. Consequently every bad displayed flag lies in an earlier true Harder--Narasimhan type. Delete the union of the earlier proper images and call the resulting open $U_\tau$. Its receiver has exactly the good decreasing semistable flags. Conversely, a good flag is the unique Harder--Narasimhan flag. This identifies the open substacks, since an open substack is determined by its geometric points. Evaluation is now proper and universally injective. Its relative tangent is filtered by the forbidden $\Hom$ groups between decreasing semistable factors, and these vanish. It follows that evaluation is a representable closed immersion \begin{equation}\label{cmp:closed-hn} i_\tau:S_\tau\into U_\tau. \end{equation} This also gives the unique relative flag over nonreduced bases. Use $U_\tau$ also for the derived open of the physical window with this classical truncation. Choose ordered integral weights for this flag type, with weight zero on its unique framed graded piece. Let $G_\tau$ be the open product of its derived good graded-factor stacks. Using \eqref{phys:filtered-stacks}, define \[ \mathfrak F_\tau=\operatorname{Filt}(U_\tau)_{\tau} \mathop{\times}_{\operatorname{Grad}(U_\tau)}G_\tau, \qquad G_\tau\xleftarrow{q}\mathfrak F_\tau\xrightarrow{p}U_\tau. \] The framing at infinity stays fixed in this mapping-stack problem. Proposition~\ref{phys:critical-equivalence}, applied to universal finite exact diagrams, identifies its truncation with the coherent flag rows and compatible fields constructed above. Every extension of the good factors lies in the cutoff window by Lemma~\ref{cmp:cyclic-bound}. Its decreasing semistable flag is its Harder--Narasimhan flag, so it lies in the current open $U_\tau$. Conversely a filtered object with these good graded terms has that coherent flag. These are open conditions on the common exact-diagram problem, determined by geometric fibers; hence the identification holds over derived test bases. The Rees construction realizes the chosen weights, and its classical evaluation is $t_0(p)=i_\tau$ in \eqref{cmp:closed-hn}. Let $K_{G_\tau}$ denote the product DT coefficient with the moving weight normalization in integral localization. The oriented localization map of Proposition~\ref{phys:integral-localization}, equivalently \cite[Corollary 7.19]{KPS} with its monodromic Hodge construction, is \begin{equation}\label{cmp:zeta} \zeta:K_{G_\tau}\xrightarrow{\sim}Rq_*p^!\Phi_{U_\tau}. \end{equation} The good product orientation is the one obtained from the same physical orientation; its moving determinant and all refinement maps are those of \cite[Proposition 8.34]{KPS}. \begin{lemma}\label{cmp:counit} Let $h_{U_\tau}$ and $h_{G_\tau}$ be the spectral maps to $\cA_r^L$, using spectral addition on the graded factors of $G_\tau$. After Hitchin direct image, the induction map of the good factor coefficient is the closed-stratum costalk counit \begin{equation}\label{cmp:costalk} Rh_{U_\tau,*}i_{\tau,*}i_\tau^!\Phi_{U_\tau} \longrightarrow Rh_{U_\tau,*}\Phi_{U_\tau}. \end{equation} Its cofiber is the direct image from the next open. \end{lemma} \begin{proof} Let $c:q^*K_{G_\tau}\to p^!\Phi_{U_\tau}$ be the adjoint of $\zeta$. The triangular identity says exactly that $Rq_*c\circ\eta_q=\zeta$. Since $h_{U_\tau}p=h_{G_\tau}q$, the integrated induction map is \[ Rh_{G_\tau,*}K_{G_\tau} \xrightarrow{Rh_{G_\tau,*}\zeta} R(h_{U_\tau}p)_*p^!\Phi_{U_\tau} \xrightarrow{Rh_{U_\tau,*}\epsilon_p} Rh_{U_\tau,*}\Phi_{U_\tau}. \] Mixed Hodge modules are insensitive to derived nilpotent thickenings of the underlying map. With $t_0(p)=i_\tau$, its second arrow is \eqref{cmp:costalk}. The localization triangle for the complementary open $j$ has cofiber $Rh_{U_\tau,*}Rj_*j^*\Phi_{U_\tau}$. Products, refinements, and restrictions use these same units and counits, so their comparison cells are retained. \end{proof} In the split row, apply $\phi_{g_{\cA}}$ after these maps have been constructed. The trace potential is additive under finite spectral addition. Moreover the split coefficients are equivariant for a normal dilation under which $g_{\cA}$ has weight two. After the degree-two Kummer cover they are locally acyclic in the function direction away from zero. Thus the relevant vanishing cycles of a sum are supported where every summand has value zero. Proper exchange for finite addition and monodromic Thom--Sebastiani give exactly the lower-rank source images used in $\cI_r$. \subsection{Vacuum removal and finite wall transport} For $i=0,1$, let $\mathscr X_{i,r,\chi}^{\times}(c)$ be the physical nonzero-section window in the row $X_i$, with its map $h_i$ to $\cA_r^L$. Set \[ W_{i,r,\chi}^{\times}(c)=R(h_i)_* \bigl(\Phi_{\mathscr X_{i,r,\chi}^{\times}(c)}\otimes\La^\chi\bigr). \] We suppress $i$ in assertions proved separately in both rows. Let $R_{r,\chi}$ denote the corresponding stable-pair restriction, with the same coefficient $\Phi\La^\chi$. \begin{proposition}\label{cmp:restrictions} For every $c\leq c_r$, the restriction to the stable-pair chamber has cofiber in $D_{\cI_r}$. The same is true for restriction to every generic $t>\max(\chi/r,0)$ and for the nested-window restriction \begin{equation}\label{cmp:rho} \rho_\chi(c_1,c_2): W_{r,\chi}^{\times}(c_1)\longrightarrow W_{r,\chi}^{\times}(c_2),\qquad c_1\max(\chi/r,0)$, the wall stack has no $s=0$ object: its vacuum subobject would have slope $\lambda$ and its unframed quotient slope $\chi/r$. A rank-zero framed factor would be the vacuum and would force $\lambda=\chi/r$. Therefore both ranks are strictly between $0$ and $r$, so all closed terms lie in $D_{\cI_r}$. The octahedral triangle transports \eqref{cmp:large-vacuum} through the finitely many walls between $t$ and $T$. Two cutoffs $c_10\qquad(1\leq ec_r$. This proves containment for both complete intrinsic chamber stacks. \end{proof} \begin{lemma}\label{cmp:endpoints} The core surface window is empty for $\chirM_r$. For the ordinary source modulo $\cS_r$, the upper vanishing endpoint is instead $\chi>-a_r$. \end{lemma} \begin{proof} Since $\ell$ is even, $\alpha_r$ is an integer. A rational slope of denominator at most $r$ below $\alpha_r$ is at most $\alpha_r-1/r$. Thus $\mu_{\min}>c_r$ is equivalent to $\mu_{\min}\geq\alpha_r$. If $\chih+(r-1)\ell$. Increase this bound to the projective framing threshold. Consider a semistable physical object in the immediate chamber $t=\chi/r+\epsilon$, $0<\epsilon<1/r^2$. If its projected Higgs bundle $E$ were unstable, its highest Higgs Harder--Narasimhan factor $H$ would satisfy $\mu(H)>\chi/r$. The denominators are at most $r$, so $\mu(H)>t$. The preceding bound, applied to the semistable bundle $H$, gives $H^1(C,H)=0$. Serre duality then gives $\varepsilon|_H=0$ for the dual framing $\varepsilon:E\to K_C$. The moment relation makes the off-diagonal transverse field $H\to(E/H)A$ a Higgs morphism. In the nonsplit row the local transverse lifts differ by a multiple of $\Theta$, so their off-diagonal blocks agree. Negativity of $A$ and the decreasing Higgs slopes force this morphism to vanish. Thus $H$ is an unframed physical subobject of slope greater than $t$, a contradiction. Consequently $E$ is semistable, $H^1(C,E)=0$, and $\varepsilon=0$. If $s=0$, the vacuum subobject and its full unframed quotient would give $t\leq\chi/r$, a contradiction. Thus $s\neq0$. The same negative-$A$ Hom vanishing preserves every equal-slope subobject. The remaining framed stability condition is the Joyce--Song condition: the nonzero section is contained in no proper Higgs subbundle of the same slope. Denote this pair space by $\cP^{\mathrm{JS}}$. The ordinary pair stack is therefore the smooth stable framed open, projective over the positive semistable Hitchin moduli; the projective framing construction is described in \cite[Definition 2.11 and (2.37)]{PToda}. The relative framing map is smooth because $H^1(E)=0$, and the positive Higgs stack is smooth. The framed stability kills stabilizers. Thus its Hitchin map is proper and its two critical presentations are $T^*[-1]\cP^{\mathrm{JS}}$ and $\Crit(\cP^{\mathrm{JS}},g)$ with the orientations of Section~\ref{sec:physical}. On this open $\delta$ and the proper exchange $e_h$ are isomorphisms. Proposition~\ref{cmp:restrictions} then gives the upper endpoint for $K_\chi$. The ordinary-source endpoint is needed only in the split row. Use the full source, with the zero section retained, and work modulo $\cS_r$. Its restriction to a sufficiently large positive chamber is an isomorphism in this quotient by Proposition~\ref{cmp:restrictions}. The same proper-receiver argument transports it across every wall between that chamber and $-\epsilon$, where $0<\epsilon<1/r$. Here one permits the rank-zero framed factor $\OO$ and its single rank-$r$ unframed complement. The latter has allowed support by the independent Proposition~\ref{source:unframed-support}; every other framed factor has smaller rank and is covered by the source induction. At the zero wall the purity argument is unchanged: if the unframed quotient has average zero, its nonnegative finite slopes and its torsion length force all slopes and that length to vanish. All walls in this bounded interval have the finite numerical lists and complete coherent receivers constructed above. The transport uses the independent rank-$r$ unframed support theorem and the smaller-rank source induction. Lemma~\ref{cmp:split-reflection} places both immediate chambers in the core and identifies their normalized direct images by \eqref{cmp:normalized-reflection}. If $\chi>-a_r$, then $-\chi0$, its special fiber vanishes on $D$ and contradicts \eqref{cmp:full-frame}; hence $m=0$. Its determinant is a section of a degree-zero line, nonzero on $D$, and therefore nowhere zero on the special fiber. It is an isomorphism, and uniqueness follows from torsion-freeness over the DVR. Fields and sections extend compatibly for the same reason. Universally, the diagonal can also be described as the zero locus of the boundary obstruction of the prescribed frame in $R^1\pi_*\mathcal Hom(E,F)(-D)$; the vanishing of $R^0$ makes the extension unique. This proves the closed-diagonal statement over arbitrary bases and also removes stabilizers. In the framed product of the initial level-two and final level-zero endpoints over $T$, take the scheme-theoretic closed union $\widetilde Z$ of the two word images and the marked graph $\widetilde{\mathfrak D}$. A word is proper over its final endpoint; its map to the product is proper because the initial endpoint is separated. Each image, their union, and the graph are therefore proper over the final endpoint. Quotient by the common diagonal frame group $G=\operatorname{Res}_{D/T}\GL_r$: \[ Z=[\widetilde Z/G],\qquad \mathfrak D=[\widetilde{\mathfrak D}/G]. \] The right map $b:Z\to X_{\chi,0,T}$ is representable proper, as is checked after the final frame atlas. The parameter $T$ is retained in both endpoints and in this proper pushforward. The intrinsic maps and homotopies will descend along $T\to C\times C$ before the proper curve trace is applied. The graph $\mathfrak D$ is smooth of dimension $q=d_\chi+\dim T-1$. Smooth descent applies to $G$, including its unipotent part. The equalities of functions and residue coordinates descend to the scheme-theoretic images and their union. If $a$ is the left map, then on the carrier itself \begin{equation}\label{cmp:carrier-identities} a^*F_{\mathrm{initial}}=b^*F_{\mathrm{final}}, \qquad a^{-1}S_{\mathrm{initial}} =b^{-1}S_{\mathrm{final}} \end{equation} as functions and closed schemes. For ordinary Hodge Borel--Moore theory use the mapping spectrum \[ \operatorname{BM}_q(Z)= \operatorname{Map}(\Q_Z,\omega_Z\LL^q). \] Each derived word is a derived base change of the regular diagonal of the smooth framed middle endpoint. Both are quasi-smooth of virtual dimension $q$. The fundamental class is the coordinate-class specialization, localization boundary, and inverse Thom map; its composition and base-change paths are those of \cite[\S2, Constructions 3.1, 3.4, 3.6 and Theorems 3.12--3.13]{Khan}. In this regular-diagonal situation the binary path is also the double-deformation diagram of \cite[Sections 3.2.19--3.2.21]{DJK}. Swapping the two odd coordinate classes contributes $-1$, and swapping the two odd localization boundaries contributes $-1$. They cancel. This fixes the binary path, including its sign. Transport the class by the nil-invariant equivalence induced by $t_0W\to W$ and properly push the classical truncation to $Z$. The bicartesian equivalence above identifies the two classes off $\mathfrak D$, with these binary paths. Localization lifts their difference to $\operatorname{BM}_q(\mathfrak D)$. Smooth purity identifies its underlying mapping space with \[ \operatorname{Map}(\Q,R\Gamma(\mathfrak D,\Q)). \] There is no negative cohomology, so the space is discrete with components $H^0(\mathfrak D,\Q)$, one scalar per connected component. This reduction takes place in Hodge theory. On the dense source open where $\Theta(y)$ is regular semisimple and every eigencomponent of $s(y)$ is nonzero, removal first is impossible. Creation followed by removal has one branch for each eigenline. On its $j$th chart the condition is exactly \begin{equation}\label{cmp:eigenline-equation} (y-x)s_j(y)=0. \end{equation} The second factor is a unit, so the branch is smooth of dimension $q$. Its quasi-smooth virtual dimension is also $q$; hence the obstruction space vanishes there, and its fundamental class has complex multiplicity $+1$. Evaluation surjectivity for the meromorphic bundles makes this open meet every component of the smooth graph. The eigenline cover has degree $r$, and its trace is $r$. We conclude \begin{equation}\label{cmp:bm-relation} [W_+]-[W_-]=r[\mathfrak D] \quad\hbox{in }\pi_0\operatorname{BM}_q(Z), \end{equation} with the specified binary composition paths. The localization lift is locally constant, so the dense calculation determines it. \subsection{Lifting the class to the entire arrow} Let $X\xleftarrow a Z\xrightarrow bY$ be such a carrier with $b$ proper and $X,Y$ smooth. If $d=\dim X$, dualizing purity and proper adjunction give \begin{align}\label{cmp:bm-map} \operatorname{BM}_q(Z) &\simeq\operatorname{Map}(a^*\omega_X, b^!\omega_Y\LL^{q-d})\nonumber\\ &\simeq\operatorname{Map}(b_*a^*\omega_X, \omega_Y\LL^{q-d}). \end{align} Pushing a class properly inserts the pullback unit followed by the proper counit. Thus \eqref{cmp:bm-map} represents the original convolution maps and their binary paths. Let $i_X:S_X\into X$ and $i_Y:S_Y\into Y$ be the residue-zero inclusions, and put $\Gamma_X=i_{X,*}i_X^!$ and $\Gamma_Y=i_{Y,*}i_Y^!$. Write $K_X,K_Y$ for the ambient dualizing normalizations, absorbing the indicated Tate factors. Put $A_0=b_*a^*\Gamma_XK_X$ and $A_1=b_*a^*K_X$. By \eqref{cmp:carrier-identities}, $A_0$ is supported on $S_Y$. The mapping spectrum of squares to $\Gamma_YK_Y\to K_Y$ is \begin{equation}\label{cmp:arrow-coreflector} \operatorname{Map}(A_0,\Gamma_YK_Y) \mathop{\times}\limits^h_{\operatorname{Map}(A_0,K_Y)} \operatorname{Map}(A_1,K_Y). \end{equation} The first map to the base is an equivalence, since $\Gamma_Y$ is the coreflector to supported objects. Consequently \eqref{cmp:arrow-coreflector} projects equivalently to $\operatorname{Map}(A_1,K_Y)$. Its inverse produces the supported map and its square homotopy with contractible choice. It is additive and preserves the binary composition paths. Write $\phi_X,\phi_Y,\phi_Z$ for vanishing cycles of the common function on the three spaces, and $a_0,b_0$ for the restrictions of $a,b$ to its zero fibers. Apply vanishing cycles to the entire square. Its source is $\phi_X\Gamma_XK_X$. The critical correspondence uses \[ b_{0,*}a_0^*\phi_XK_X\longrightarrow b_{0,*}\phi_Za^*K_X\xrightarrow{\sim} \phi_Yb_*a^*K_X\longrightarrow\phi_YK_Y. \] The first exchange is used only in its forward direction; only the proper exchange along $b$ is inverted. In the residue presentation, let $V$ be the smooth ambient, $S=V|_{t_0\cP}$, $i:S\into V$, and $p:S\to t_0\cP$ the dual-residue vector bundle. Use $\mathscr D_\chi=\omega_V\LL^\chi$. The supported arrow is \begin{equation}\label{cmp:ambient-counit} \phi_F(i_*i^!\mathscr D_\chi)\longrightarrow\phi_F\mathscr D_\chi. \end{equation} Both vertices are supported on $S$, because derivatives in the dual residue variables are the residue equations. The source identification uses \[ i^!\omega_V=\omega_S=p^*\omega_{t_0\cP}\LL^{-e}, \qquad\phi_{p^*g}p^*\simeq p^*\phi_g, \qquad\id\simeq Rp_*p^*. \] If $d_0=B_r+2e$, its pushdown is \begin{equation}\label{cmp:counit-source-normalization} Rp_*\phi_{p^*g}i^!\mathscr D_\chi =\phi_g\omega_{t_0\cP}\LL^{\chi-e} =\phi_g\Omega_{\cP}\LL^{\chi/2}\LL^{-d_0/2}. \end{equation} Its target is $R\pi_{g,*}\Phi_{\Crit(\cP,g)} \LL^{\chi/2}\LL^{-d_0/2}$. Thus \eqref{cmp:ambient-counit} is the local $\delta$ in \eqref{cmp:delta}, with the same common factor at both vertices. Closed base change identifies its supported correspondence with the ordinary $\Omega_{\cP}$ correspondence after this vector-bundle pushdown. Finally the nonproper exchange $e_h$ is the right mate of the forward pullback exchange. If $h:X\to B$ and $h_0:X_0\to B_0$ are the maps before and after the zero fiber, then on $K$ the mate is \[ \phi_Bh_*K\longrightarrow h_{0,*}h_0^*\phi_Bh_*K \xrightarrow{\operatorname{Ex}_h} h_{0,*}\phi_Xh^*h_*K\longrightarrow h_{0,*}\phi_XK. \] The outer maps are the pullback unit and counit. Composition of forward exchanges and the adjunction triangle identities determine its composition homotopies. For the ordinary action of a correspondence $u$, use those same homotopies in \[ T(u):h_{X,*}K_X\xrightarrow{\eta_a} (h_Xa)_*a^*K_X=h_{Y,*}b_*a^*K_X \xrightarrow{u}h_{Y,*}K_Y, \] its mate identities give \[ e_{h_Xa}=(h_{X,*}e_a)(e_{h_X}a_*),\qquad e_a\phi_X(\eta_a)=a_*\operatorname{Ex}_a(\eta_a\phi_X). \] They identify the square for $e_h$ with the square already obtained for the local $\delta$. The complete diagram has rows \[ \phi_{g_{\cA}}Rh_*\Omega\longrightarrow Rh_*\phi_g\Omega\longrightarrow R(h\pi_g)_*\Phi, \] and includes the two square homotopies. Composing the horizontal arrows is an exact functor on enhanced diagram categories. It takes \eqref{cmp:bm-relation} to a relation on precisely $\Delta$, and then on its cofiber. \subsection{Independence of the auxiliary presentation} Enlarging the divisor adds an acyclic equation-variable block $[W\xrightarrow d E]$. On a splitting torsor, a triangular change of equations and its dual reduce the added potential to $\langle d(c),\zeta\rangle$ with support $c=0$. Differences between extensions $\beta_D$ belong to the residue ideal: the principal-part expansion of $\Tr\Theta^2$ expresses them as linear combinations of the residue coordinates. Translating the corresponding dual variables removes them. For one hyperbolic pair write $c=z_2+\mathrm i z_1$, $\zeta=z_2-\mathrm i z_1$. Then $c\zeta=z_1^2+z_2^2$. Projection of the positive real thimble to $\zeta$ preserves orientation, whereas projection to $c$ reverses it. Thus the raw support-counit scalar for a rank-$k$ block in the compact dimensional-reduction convention is $(-1)^k$. Equivalently this is $\binom{2k}{2}\equiv k\pmod2$ in the duality normalization of \cite[Lemma 3.4]{Kinjo}. The grouped orientation form is \[ (-1)^{k(k-1)/2}(-\mathrm i/2)^k dc_1\cdots dc_k\,d\zeta_1\cdots d\zeta_k. \] For invertible $d$, the contraction root contains $\det d$; its square is the Hessian determinant with this same parity. Changes of bundle frame act inversely on the dual half, so the total determinant is one. In our presentations $e=mr(r+1)$ and every enlargement rank are even. Hence the raw stabilization scalar is $+1$, and the existing local $\delta$ and its binary paths are preserved. Splittings and potential extensions form affine torsors. On the fixed intrinsic endpoint objects their pullback is fully faithful, by $\id\simeq Rp_*p^*$ and the projection formula. The preceding transformations therefore descend the maps and their homotopies. To compare a previous one-point divisor to the two-point family, use their sum, which need only avoid the active mark of that one operator. Descend the one-step comparison first, then compose on the whole $C\times C$. Changing bounded bundle opens is handled by supported open excision of \eqref{cmp:ambient-counit}. Both vertices are supported on the complete residue-zero window. Restriction to an open containing it is fully faithful on the objects and their mapping spaces. On a Hecke diagram first restrict the proper target, then take its full inverse image. If a further source excision is needed, delete the proper image of its bad closed inverse image; that image misses the target residue support. This gives a common refinement preserving the proper arm, fundamental class, coreflector square, and binary path. The three directed presentations consequently define the same intrinsic arrows and restriction squares. \begin{proposition}\label{cmp:weyl-relation} There are degree-zero creation and removal morphisms on the arrows $\Sigma_\chi(c)$, \[ E_\chi^c:\Sigma_\chi(c-1)\longrightarrow\Sigma_{\chi+1}(c), \qquad F_\chi^c:\Sigma_{\chi+1}(c-1)\longrightarrow\Sigma_\chi(c), \] natural for the cutoff restrictions, with relation \begin{equation}\label{cmp:weyl-buffer} F_\chi^c E_\chi^{c-1} -E_{\chi-1}^c F_{\chi-1}^{c-1} =r\rho_\chi(c-2,c). \end{equation} The same holds on $K_\chi(c)$ and on the normalized full ordinary source \begin{equation}\label{cmp:ordinary-source} \mathcal A_\chi(c)=Rh_* \omega_{t_0\cP_{r,\chi}^{\mathrm{all}}(c)} \LL^{\chi+B_r/2}, \end{equation} where the zero section is allowed. \end{proposition} \begin{proof} At a fixed mark creation has target $\Sigma_{\chi+1}\LL^{-1}$; removal already has degree zero. Indeed the ambient dimensions in \eqref{cmp:ambient} and \eqref{cmp:bm-map}, together with the common factor $\LL^{-d_0/2}$ in \eqref{cmp:counit-source-normalization}, give these twists. The class relation on $C_x\times C_y$ is therefore \[ f_\chi^c(y)e_\chi^{c-1}(x) -e_{\chi-1}^c(x)f_{\chi-1}^{c-1}(y) =r\rho_\chi(c-2,c)\boxtimes\operatorname{cl}(\Delta_C). \] Fix $y$ and integrate the moving mark $x$ by the proper trace of the whole curve $R\Gamma(C,\Q)\to\LL$. It cancels $\LL^{-1}$ and sends $\operatorname{cl}(y)$ to $1$. This gives \eqref{cmp:weyl-buffer}. The graph term is the inclusion of the level-zero endpoint into level two, so its action is the restriction $\rho$. Naturality follows by proper target base change and the preceding supported excision. The enhanced cofiber functor gives the same relation on $K_\chi$. For the ordinary source use the same smooth meromorphic ambient with $s=0$ allowed. The regular-semisimple open used in \eqref{cmp:eigenline-equation} is still dense in every component. Apply the coreflector but not vanishing cycles. Its pushed coefficient is $\omega_{t_0\cP}\LL^{\chi-e}$, differing from \eqref{cmp:ordinary-source} by the single common factor $\LL^{-B_r/2-e}$. This cancels on every edge. The same trace gives the ordinary-source relation, without requiring the holomorphic pair stack itself to be smooth. \end{proof} \subsection{Finite-band vanishing} \begin{lemma}\label{cmp:weyl-nilpotence} Let $V=\bigoplus_{a\leq\chi\leq b}V_\chi$ in a $\Q$-linear abelian category. If degree-zero morphisms $E,F$ raise and lower $\chi$ and satisfy $[F,E]=r\id$ for an integer $r>0$, then $V=0$. \end{lemma} \begin{proof} $E$ is nilpotent, and $[F,E^m]=\sum_{j=0}^{m-1}E^j[F,E]E^{m-1-j} =mrE^{m-1}$. Starting with $E^m=0$ and decreasing $m$ gives $\id=0$. \end{proof} \begin{theorem}\label{cmp:comparison} For every $r,\chi$ and $c\leq c_r$, the cofiber of the original comparison arrow \eqref{cmp:delta} belongs to $D_{\cI_r}$. \end{theorem} \begin{proof} Fix a perverse degree $j$, and let $q_r:\MMHM(\cA_r^L)\to\MMHM(\cA_r^L)/\cI_r$ denote the Serre-quotient functor on the perverse heart. Proposition~\ref{cmp:restrictions} makes every cutoff restriction an isomorphism. Use its unique inverse to transport the operators in Proposition~\ref{cmp:weyl-relation} to \[ V_\chi=q_r\pH^jK_\chi(c_r). \] Restriction naturality makes their relation $[F,E]=r\id$. Lemma~\ref{cmp:endpoints} leaves a finite characteristic band, independent of $j$. Lemma~\ref{cmp:weyl-nilpotence} gives $V_\chi=0$ for all $\chi$. All cutoffs follow by their actual restrictions. Only the finite band and its adjacent values were needed for the three directed bundle levels; iteration of $E$ now occurs inside the quotient category. \end{proof} \begin{theorem}\label{cmp:source-support} Every perverse constituent of the normalized split stable-pair direct image belongs to $\cS_r$. \end{theorem} \begin{proof} We prove by induction on $r$ the support assertion for every full ordinary window $\mathcal A_\chi(c)$, $c\leq c_r$, and for its complete chamber and wall coefficients with $t>-1/r$. The rank-zero framed factor is the vacuum. The positive unframed factors have allowed supports by Proposition~\ref{source:unframed-support}. In the full-source Harder--Narasimhan restrictions, every smaller framed-rank factor has allowed supports by this induction. For $e-1/r$ is contained in $t>-1/e$; nested cutoff restriction compares a lower-rank factor at cutoff $c$ with its own core. Finite spectral addition sends the concatenated partition supports to the corresponding $\Sigma_\nu$. The only one-factor rank-$r$ unframed term is exactly the independent positive input. Thus source restrictions are isomorphisms in the quotient by $\cS_r$. The lower empty endpoint and the source reflection of Lemma~\ref{cmp:endpoints} leave the band $[a_r,-a_r]$. Apply the ordinary-source relation of Proposition~\ref{cmp:weyl-relation} and Lemma~\ref{cmp:weyl-nilpotence} in each perverse degree. This proves support for the full core source, and nested restriction gives it for every $c\leq c_r$. For $t>-1/r$, the slope-denominator argument of Lemma~\ref{cmp:split-reflection} and the cyclic bound place the complete chamber and wall stacks inside the core. Their restrictions from the full window are successive closed-stratum costalk triangles from the proper-receiver wall argument in the proof of Proposition~\ref{cmp:restrictions}. Each closed term has either a smaller positive framed rank, or the framed vacuum and a rank-$r$ unframed factor. The former is covered by the induction hypothesis; the latter by Proposition~\ref{source:unframed-support}. Finite addition and the Serre property place every closed term in $D_{\cS_r}$. The triangles therefore give support for these chamber and wall coefficients, completing the induction. Dimensional reduction identifies the ordinary source \eqref{cmp:ordinary-source} with the split physical coefficient. Taking a sufficiently large positive chamber gives the stable-pair assertion. \end{proof} \begin{corollary}\label{cmp:pt-difference} Let $R_{0,r,\chi}$ and $R_{1,r,\chi}$ be the normalized stable-pair direct images from $X_0$ and $X_1$. Then \begin{equation}\label{cmp:bounded-pt-difference} [R_{1,r,\chi}]-[\phi_{g_{\cA}}R_{0,r,\chi}] \in K_0(\cI_r). \end{equation} \end{corollary} \begin{proof} The available morphisms are the original window comparison $\phi W_0\to W_1$ and the two restrictions $\phi W_0\to\phi R_0$, $W_1\to R_1$. Their cofibers lie in $D_{\cI_r}$ by Theorem~\ref{cmp:comparison} and Proposition~\ref{cmp:restrictions}. They form a zigzag of isomorphisms in each perverse quotient. The stable-pair endpoints are bounded proper direct images by Proposition~\ref{fock:properness}. Taking their Grothendieck classes and applying the Serre localization sequence gives \eqref{cmp:bounded-pt-difference}. \end{proof} \section{The positive source identity} \label{sec:source} We first compute the split stable-pair coefficient on $X_0$. Let $h_{0,r,\chi}:P_{r,\chi}(X_0)\to\cA_r^L$ be the positive spectral map, and set \begin{equation}\label{source:R-L} R^L_{r,\chi}=R(h_{0,r,\chi})_* (\Phi_{P_{r,\chi}(X_0)}\otimes\La^\chi). \end{equation} Dimensional reduction and the ordinary-source argument of Theorem~\ref{cmp:source-support} put every perverse cohomology object of \eqref{source:R-L} in $\cS_r$. Choose any Euler characteristic $\chi_*$ coprime to $r$. Write $M_L(r,\chi_*)$ for the smooth stable positive Higgs moduli space, and let $h_L:M_L(r,\chi_*)\to\cA_r^L$ be its Hitchin map. Put \begin{equation}\label{source:positive-perverse} P_i^L=\pH^iR(h_L)_*\Q_{M_L(r,\chi_*)}[B_r+1]. \end{equation} This Hodge module is independent of $\chi_*$ by \cite[Theorem 0.4]{MSChi}. Define \begin{equation}\label{source:b-gamma} b_r=\frac{\ell r(r+1)}2-rh,\qquad \gamma_r=1+rh+\frac{\ell r(r-1)}2. \end{equation} Thus $\dim\cA_r^L=b_r$, $\gamma_r$ is the arithmetic genus of a rank-$r$ spectral curve, and $b_r+\gamma_r=B_r+1$. Let $x$ be the formal variable recording Euler characteristic. \begin{theorem}\label{source:positive-formula} The complexes $R^L_{r,\chi}$ are ordinary pure Hodge complexes of weight zero. In the coefficientwise completed Grothendieck group of mixed Hodge modules on the positive Hitchin bases, \begin{equation}\label{source:positive-series} \begin{split} 1+\sum_{r\geq1,\chi}Q^rx^\chi[R^L_{r,\chi}] =\Exp_{\operatorname{add}}\!\left( \sum_{r\geq1}Q^r\sum_i\sum_{u,v\geq0} x^{i+1+u+v} [P_i^L[-i-1-2v]\otimes\TT^{v-B_r/2}] \right). \end{split} \end{equation} Only $u,v$ are constrained to be nonnegative; $i$ runs through the finite perverse range in \eqref{source:positive-perverse}. The exponential includes the derived symmetric-group actions. \end{theorem} \subsection{The unframed input and its hypotheses} We begin with the unframed support statement used in Section~\ref{sec:comparison}. \begin{proposition}\label{source:unframed-support} For any fixed Euler slope, every individual perverse direct-image coefficient of the split semistable unframed moduli stack has strict supports among the spectral-addition images $\Sigma_\nu$. The rank-$r$ primitive is the positive coarse intersection complex, whose proper Hitchin direct image is independent of the Euler characteristic as an object of $D^b\MHM(\cA_r^L)$. \end{proposition} \begin{proof} Fix an Euler slope $\mu$ and let \[ \mathfrak N_\mu=\{0\}\sqcup \coprod_{r>0,\,\chi/r=\mu} \mathfrak M_L^{\ss}(r,\chi). \] We apply the smooth linear-stack integrality theorem \cite[Theorem 10.2.7]{BDIKP} to this stack and verify its hypotheses. The objects are compactly supported pure sheaves on $S=\Tot_C(L)$, or equivalently $L$-Higgs bundles on $C$. Semistable objects of slope $\mu$ form an abelian category. For example, if the image of a morphism between two such objects were not saturated, saturation would strictly raise its slope, while the quotient inequality for the source gives slope at least $\mu$. This contradicts semistability of the target. Kernels, images, cokernels, and extensions therefore remain in the same category. Direct sum and the scalar action are its coherent operations. For two objects $E,F$ in this category, the deformation complex is \[ R\Gamma\bigl(C,[\mathcal Hom(E,F) \longrightarrow\mathcal Hom(E,F)\otimes L]\bigr). \] Riemann--Roch and surface duality give \begin{equation}\label{source:euler-ext} \chi_S(E,F)=-\ell\rk(E)\rk(F),\qquad \Ext_S^2(E,F)=\Hom_S(F,E\otimes A)^\vee=0. \end{equation} The second equality uses $\deg A<0$ and semistability at the same slope. Hence $\mathfrak N_\mu$ is smooth and its numerical Euler pairing is symmetric. We also check the stronger pointwise orthogonality condition. At a closed polystable point write $E=\bigoplus_j S_j\otimes V_j$, where the $S_j$ are pairwise nonisomorphic stable objects of ranks $r_j$. Then \[ \dim\Ext_S^1(S_i,S_j)=\delta_{ij}+\ell r_ir_j. \] The stabilizer is $\prod_j\GL(V_j)$. The opposite summands $\Hom(V_i,V_j)\otimes\Ext^1(S_i,S_j)$ and $\Hom(V_j,V_i)\otimes\Ext^1(S_j,S_i)$ have equal multiplicity and admit an invariant symmetric cross-pairing, using the trace pairing on the two Hom spaces. For $i=j$, use trace on $\operatorname{End}(V_i)$ and any nondegenerate symmetric form on the multiplicity space. The gauge summands are orthogonal by the same trace pairing. Thus the stabilizer representation is pointwise almost orthogonal in the sense of \cite{BDIKP}. The semistable spectral Quot construction supplies a good moduli space $p:\mathfrak N_\mu\to M_\mu$, with affine diagonal. At fixed $(r,\chi)$ the stack is of finite type: consecutive ordinary Harder--Narasimhan slopes of a semistable $L$-Higgs bundle differ by at most $\ell$, which bounds the underlying bundles. Direct sum is quasi-compact. Indeed a decomposition over a fixed family is a tuple of complementary idempotents in its relative endomorphism space; the idempotent equations are closed, and at a fixed target rank there are only finitely many numerical decompositions. For each fixed finite character-support face, the graded-object stack is a finite product of these eigensheaf stacks. Its total-object morphism is therefore quasi-compact. This is the facewise finiteness condition of \cite[\S 2.3.7]{BDIKP}. The perfect cotangent complex has finitely many weights on each such face. The theorem is locally finite in the numerical components. We keep the union over all ranks when applying it, so that the stack remains closed under direct sum. The torus Chern parameters can be chosen on the stack itself. For $p_0\in C$, the line $\det(E|_{p_0})$ has scalar weight $r$; its rational Chern class divided by $r$ restricts to the generator of $H^2(B\Gm,\Q)$. On a graded stack the same construction for each eigensheaf gives a basis of torus parameters. Finite automorphisms of the face permute these lines. By \cite[Proposition 2.9]{MSChi}, the rank-$r$ stack is smooth irreducible of dimension $B_r$ and its coarse space is irreducible of dimension $B_r+1$, for every $\chi$. On the dense smooth spectral locus, the stack is the relative Picard stack and has exactly the scalar stabilizer. Scalar rigidification is generically an isomorphism onto the coarse moduli space. The BPS identification in \cite[Corollary 7.2.4 and Theorem 10.2.7]{BDIKP} consequently identifies the primitive with the coarse intersection complex, also when $(r,\chi)$ is not coprime. Let $\eta=(r,\chi)$ run through the nonzero numerical components with $\chi/r=\mu$, let $M_\eta$ be their coarse moduli spaces, and let $\Sym_{\oplus}$ denote the symmetric-power operation formed using direct sum. With $\IC^{\vir}_Y=\IC_Y\otimes\Th^{-\dim Y}$, the theorem gives the cohomological induction isomorphism \begin{equation}\label{source:cohi} \Sym_{\oplus}\!\left( \bigoplus_{\eta\neq0}\IC^{\vir}_{M_\eta} \otimes H^*(B\Gm)_{\vir}\right) \xrightarrow{\sim} Rp_*\IC^{\vir}_{\mathfrak N_\mu}, \quad H^*(B\Gm)_{\vir} =\La\otimes\bigoplus_{v\geq0}\TT^v[-2v]. \end{equation} It is formed from the lowest-perverse primitive inclusion, the specified Chern parameters, and the smooth Hall induction maps. It supplies object-level support. The primitive normalized by scalar rigidification differs from the unnormalized first perverse coefficient by its half-Tate factor. In the split physical threefold the unframed critical stack is $T^*[-1]\mathfrak N_\mu$. Negativity in \eqref{source:euler-ext} makes the ordinary transverse field zero; the derived statement follows from the physical model in Section~\ref{sec:physical}. Its half/dual-half triangle identifies the fixed orientation with the cotangent orientation. Numerical symmetry and \cite[Lemma 10.2.9]{BDIKP} make this orientation commutative. Dimensional reduction gives \[ \pi_*\Phi_{T^*[-1]\mathfrak N_\mu} =\omega_{\mathfrak N_\mu}\otimes\LL^{B_r/2} =\IC^{\vir}_{\mathfrak N_\mu}. \] Since $\mathfrak N_\mu$ is smooth, this is the zero-function critical chart. The critical and smooth induction maps agree by \cite[\S\S 6.2.3 and 8.1.7]{BDIKP}. The coarse Hitchin maps are proper. Spectral addition is finite, as proved below, and coarse direct sum is finite as well: its composite with the Hitchin map is proper, while a polystable object has only finitely many allocations of its Jordan--H\"older multiplicities among a fixed list of summands. Properness and quasi-finiteness give finiteness. Finally \cite[Theorem 0.4]{MSChi} identifies the positive coarse Hitchin direct image of $\IC$ with the full-support intermediate extensions of the exterior-power spectral variations, independently of $\chi$, in $D^b\MHM$. Applying the proper Hitchin pushforward to \eqref{source:cohi} therefore gives precisely the supports $\Sigma_\nu$. At each fixed perverse degree only finitely many degrees of the torus factors contribute. The single rank-$r$ unframed coefficient belongs to $\cS_r$. \end{proof} \subsection{Finite spectral addition and reduced spectral curves} \begin{lemma}\label{source:finite-addition} For every rank partition $\nu=(r_1,\ldots,r_s)$ of $r$, spectral addition $\prod_j\cA_{r_j}^L\to\cA_r^L$ is finite. Its image $\Sigma_\nu$ is irreducible and has a dense locus of reduced nodal curves with smooth irreducible components of the specified ranks. The number of cross-component nodes and the codimension of this image are both \begin{equation}\label{source:cross-nodes} e_\nu=\ell\sum_{i0$ apply $\theta$. Every target term remains nonzero after this operation: its new box lies in $\lambda$ and its height is at least $m_{ij}$. Thus the coefficient propagates to column $j-1$. At column zero the source has zero $\theta$-image whereas every target still survives; the coefficient is zero. Induction on $j$ proves $\Hom(K,M/K)^{H,R}=0$. A common fixed quotient is supported at $d[0]$ and is killed by $t^d$, by Cayley--Hamilton. It lies in a projective finite-module Quot fiber. The residual torus has a limit there, which is monomial. A nonconstant residual orbit would have positive-dimensional closure through this limit, contradicting the vanishing of its fixed tangent. This proves the classification and reducedness. Differentiating the Artin-family identification above identifies the invariant tangent with $\Ext^1(I,I)_0^{H,R}$. Stability kills the trace-free $\Ext^0$, and proper-support Calabi--Yau duality kills the invariant $\Ext^2$ and $\Ext^3$. The derived fixed tangent is therefore acyclic. For $d=0$ the same assertion follows before taking curve cohomology: the trace-free local self-Ext complex of the partition ideal has only the nonzero hook characters in \eqref{fock:hilbert-hook-tangent}. Its normalized oriented coefficient is the unit. The orientation comparisons of Section~\ref{sec:physical} give the unit for every common fixed point. \end{proof} \subsection{Closed attracting filtrations and punctual purity} We use the compact form of oriented hyperbolic localization. For a torus action $\mu$, write $i^+:Y^+\to Y$ and $p^+:Y^+\to Y^0$. With the orientation from Section~\ref{sec:physical}, it is \begin{equation}\label{fock:compact-localization} p^+_!(i^+)^*\Phi_Y\simeq\Phi_{Y^0}\otimes\La^{I_\mu}. \end{equation} Here $I_\mu$ is the index of Proposition~\ref{phys:integral-localization}, namely \[ (\rk^+H^0T_Y-\rk^-H^0T_Y) -(\rk^+H^{-1}T_Y-\rk^-H^{-1}T_Y). \] This is the localization theorem for the inverse action in the convention $p_*\eta^!$ of \cite{KPS}. On a smooth zero-potential chart the index is $d_+-d_-$. Compact contraction contributes $\La^{2d_+}$, the ambient normalization contributes $\La^{-d_0-d_+-d_-}$, and the fixed normalization is $\La^{-d_0}$. Their quotient is exactly the exponent in \eqref{fock:compact-localization}. The chart maps, orientation torsors, and their descent in $\MMHM$ were fixed in Section~\ref{sec:physical}. \begin{lemma}\label{fock:closed-filtration} Let $Y$ be a finite-type separated $\Gm$-scheme, let $V$ be an affine representation of strictly positive weights, and suppose that $Y\to V$ is proper and equivariant. If $Y$ has a linearized ample line bundle, it has a finite closed filtration whose differences are entire attracting schemes over open-and-closed unions of fixed components. This description holds on arbitrary ordinary bases. \end{lemma} \begin{proof} A power of the ample line gives a relative projective embedding. Choose homogeneous generators of the finite equivariant module defining it. Their free cover yields a closed equivariant embedding $Y\into\mathbf P(W)\times V$ for a finite-dimensional representation $W$. Order its distinct weights $w_1<\cdots0}$. Thus the sum in \eqref{fock:index-increment} is \begin{equation}\label{fock:kernel-sum} 1+2(L_{c+1}-L_c)+2\mathbf1_{c=0}. \end{equation} Counting row endpoints gives $L_c-L_{c+1}=\mathbf1_{c\in\mathcal B_\lambda}-\mathbf1_{c<0}$. Since $\alpha+j>0$, \eqref{fock:index-increment} reduces to $\Delta I=\sigma(c)$. Restricting \eqref{fock:virtual-character} to $H$ kills its quadratic term in $\mathsf Z_m$. A legal increment has difference $F_c(u)-F_c(u^{-1})$, with \[ F_c(u)=u^{-c}\bigl(1-(1-u^{-1})(1-u)\overline{\mathsf Z_\lambda}\bigr). \] The kernel $2\sgn(D)-\sgn(D-1)-\sgn(D+1)$ is supported at $D=-1,1$ with values $-1,1$. Therefore \begin{align} \Delta I_H&=\sgn(-c)+L_{c-1}-L_{c+1} =\tfrac12\bigl(\sigma(c-1)+\sigma(c)\bigr),\nonumber\\ I_H&=\tfrac12\sum_cd_c\bigl(\sigma(c-1)+\sigma(c)\bigr),\quad I_R=\tfrac12\sum_cd_c\bigl(\sigma(c)-\sigma(c-1)\bigr). \label{fock:two-indices} \end{align} The first index is constant on each graded-dimension component. Passing from compact affine cohomology to punctual cohomology changes $I_R$ to $-I_R$. After the factor $\La^a[d]$ in \eqref{fock:relative-object}, the punctual exponent and degree are \begin{equation}\label{fock:punctual-indices} J=I_H-I_R=\sum_cd_c\sigma(c-1),\qquad \text{degree}=J-d. \end{equation} We use the diagonal-trace rim-hook identity \cite[Theorem 0.2 and Section 6]{Sulzgruber}. In the present notation it asserts that reverse plane partitions are enumerated, with their individual diagonal sums, by independently repeated rim hooks. The hook of a box of content $c$, arm $a$, and leg $l$ occupies the diagonal interval $[c-l,c+a]$. That interval contains $l+1$ Maya members and $a$ nonmembers; its right endpoint is a member and the preceding left endpoint is a nonmember. Hence \begin{equation}\label{fock:hook-index} \sum_{v=c-l}^{c+a}\sigma(v-1)=l-a-1=e_b, \qquad e_b-h_b=-2(a_b+1). \end{equation} Equations \eqref{fock:punctual-indices}--\eqref{fock:hook-index} and the already proved punctual purity give the punctual graded cohomology object as the symmetric object on the generators \begin{equation}\label{fock:punctual-generators} v^{h_b}\Th^{e_b}[2(a_b+1)],\qquad b\in\lambda. \end{equation} The intrinsic weight $k+d$ separates degree $k$ before any global symmetric-power operation is taken. \subsection{Moving support and finite symmetric actions} The punctual calculation must descend with its Hodge structure and its finite permutation actions. Let $x,y$ be two \'etale coordinates on an open set $U\subset C$. On a neighborhood $V$ of the diagonal in $U_t\times U_p$, discard the other sheets of the two \'etale equivalence relations. Then \[ X=x(t)-x(p),\qquad Y=y(t)-y(p)=Xv(t,p), \] where $v$ is a unit. Over $B=U_p\times\A^1_s$ form \begin{equation}\label{fock:coordinate-dnc} W=\Spec\bigl(\OO(V)[s,\tau]/(X-s\tau)\bigr), \qquad \upsilon=\tau v(t,p). \end{equation} It has two \'etale maps to $\A^1_\tau\times B$ and $\A^1_\upsilon\times B$. At $s=1$ they are the original coordinate maps; at $s=0$ their ratio is the linear change $\upsilon=(dy/dx)(p)\tau$. Cotangent lifting preserves the relative threefold volume and the transverse monomial module. For a fixed $d$, the support-cycle fiber $d$ times the distinguished section is the constant punctual Quot scheme over $B$. This holds on nonreduced bases: formal \'etaleness and Cayley--Hamilton identify the quotients killed by the $d$th power of the section coordinate. The supported fixed-determinant gluing construction in Proposition~\ref{phys:parameter-square} applies to these two maps. On the completed parameter ring $R=\C[[p-p_0,s-s_0]]$ its reference branch is constant. The full-parameter calculation there gives equality of the two potentials modulo the square of the common critical ideal, for these same maps. Its algebraic four-Hom line map has square equal to the determinant of the same cotangent derivative. Faithful flatness of completion therefore identifies the existing algebraic oriented critical structures over $B$. Tensor the line map with $\id_{K_B}$ to retain the smooth-base factor. Proper pushforward of the punctual comparison gives a map between pullbacks of one punctual cohomology complex over $U\times\A^1$. Pullback from $U$ is fully faithful, by the affine-line cohomology adjunction. The two endpoint maps are equal. At $s=0$ this is the linear coordinate action. The physical orientation map gives an action over $\Gm$, and its induced maps on cohomology objects are morphisms of constant variations. They equal the identity at $1$, and hence everywhere on the connected group. Thus all moving coordinate transitions are identity on the punctual Hodge variations. On an ordered configuration space the finite defect decomposes canonically by support idempotents, also over nilpotent bases: $T=\bigoplus_iT_i$. Use a single baseline $G_0$ and the triangles $G_0\to G_i\to T_i$. Put \[ M(G)=\det\fib\bigl(\RHom_{S_K}(G,G) \longrightarrow R\Gamma(S_K,G)\bigr). \] Use the equivariant surface and transported orientation of Lemma~\ref{phys:supported-determinant}. The following grid is \(H\)-equivariant before taking determinants. Its weight-zero part compares the fixed orientations; its moving parts supply the index and permutation factors below. The supported Hom and section triangles give the line map \begin{equation}\label{fock:baseline-factorization} M(G)M(G_0)^{-1}\simeq \bigotimes_i\bigl(M(G_i)M(G_0)^{-1}\bigr). \end{equation} The cross complexes $\RHom(T_i,T_j)$ vanish canonically for $i\ne j$; the volume trace splits by the same idempotents. Taking weight zero in this grid before determinants gives the oriented fixed-coefficient comparison, with the determinant functor's regrouping maps. The merge morphism is \'etale on the labeled branch, since the idempotents lift uniquely over infinitesimal extensions. When two reference curves have baselines of different parity, divide first by their baseline point coefficients. The quadratic-point calibration in \cite[Lemma 3.4]{Kinjo} and \cite[Section 7.8]{KPS} is a rational unit map of the associated orientation torsors. The compact Thom and product-thimble maps have scalar $+1$. This fixes the cancellation in \eqref{fock:baseline-factorization}. The ordered-configuration variations are consequently constant; their cohomology still retains the topology of the configuration space. We check the symmetry used in their descent. In the $J$-model of monodromic Hodge modules, let $r(x)=x^2$, let $\iota(x)=-x$ be its deck involution, and put \[ \Th=e_-r_*\Q_{\A^1}[1],\qquad e_-=(1-\iota)/2. \] Its global cohomology vanishes. Its $k$th convolution power is the full-sign summand of $(\sum x_i^2)_*\Q_{\A^k}[k]$. A smooth quadratic fiber retracts equivariantly onto $S^{k-1}$; its vanishing cohomology has determinant character. A coordinate permutation has its sign on this cohomology, and the shifted $[1]$ factors have the same Koszul sign. The product is $+1$. These objects lie in the perverse heart, where rational realization is faithful. This proves that $\Th$ is even and $\La$ is odd, as stipulated in \eqref{eq:tate-conventions}. For a relative defect packet, let $m_0,m_+,m_-$ be the Euler ranks of the surface half in weights zero, positive, and negative. The four-Hom comparison gives \begin{equation}\label{fock:packet-parities} m_0+m_++m_-=-d,\qquad a=m_--m_+,\qquad \kappa=m_0+2m_+,\qquad \kappa+a=-d. \end{equation} The graded orientation exchange contributes parity $\kappa$. After the normalization $\La^a[d]=\Th^a[d-a]$, the extra exchange exponent for two packet colors is \[ \kappa_i\kappa_j+(d_i-a_i)(d_j-a_j) =2d_id_j+2a_ia_j. \] For repeated packets the remaining exponent is \begin{align*} \kappa-a^2+d^2 &=m_0(m_0+1)+2m_++2m_0(m_++m_-)+4m_+m_-. \end{align*} Both are even for all integer Euler ranks. The permutation group relations themselves come from determinant and Thom--Sebastiani coherence. The odd finite-defect correction $(-1)^d$ is multiplicative and has ratio one under exchange and refinement. Thus the finite symmetric actions are the ordinary derived actions on \eqref{fock:punctual-generators}. In particular both charge twists $F_d\mapsto F_d\otimes\Th^{-d}$ and $F_d\mapsto F_d\otimes\Th^d[-2d]$ are symmetric monoidal. Thom--Sebastiani products are formed in $\MMHM$. Choose local critical functions vanishing on their reduced critical loci and shrink to the zero-critical-value situation of \cite[Theorem 5.3]{SaitoTS}. \subsection{Global purity and proof of the relative formula} Write $A_a=Rf_{d,a,*}\Phi_{d,a}$ before twisting by its index. The global support map is proper, since the dualizing Quot scheme on the finite thickening of the projective curve is projective. At a support cycle $\sum d_ip_i$, proper base change and \eqref{fock:baseline-factorization} identify its coefficient with the external product of the punctual coefficients, summed over the compatible local indices. The punctual result shows that every stalk $H^j(i_x^*A_a)$ is pure of intrinsic weight $j$. It follows that $\wt(A_a)\leq0$. More explicitly, on a smooth stratum of dimension $s$, a lisse perverse coefficient is a variation shifted by $[s]$ and its perverse weight is its fiber weight plus $s$. Applying this to a stratification adapted to the weight filtration proves the bound. The untwisted $\Phi_{d,a}$ is self-dual, and proper pushforward preserves duality. Thus the lower weight bound also holds: \begin{equation}\label{fock:relative-purity} \wt(A_a)=0,\qquad \wt(L_{\lambda,d})=d. \end{equation} We have used self-duality before the twist by $\La^a$. Pure polarizable monodromic Hodge modules are semisimple. One can see this directly in the weight-stable Serre quotient defining $\MMHM$: a pure quotient object of weight $w$ has the pure ambient lift $\Gr^W_w M$. The latter is semisimple, and its image under an exact Serre quotient remains semisimple. This also splits the same-weight extensions used in the punctual purity argument. Stratify $\Sym^d C$ by support multiplicities. On the ordered finite cover of a stratum, the preceding coordinate and product comparisons give the constant punctual objects with their computed stabilizer actions. The right-hand side of \eqref{fock:relative-series} has the same description: it assigns box occupancies at each support point, with total weighted multiplicity $\sum h_br_b$. Taking finite rational invariants is exact. Equivariant tensor-power filtrations of short exact sequences show that passage to full $K_0$ classes preserves these projectors. For a finite group quotient of $C^s$, the invariant direct image of its constant object is the constant object of the symmetric product, including at diagonals: the unit followed by the projector is an isomorphism on the finite orbit fibers. Therefore finite closed/open localization on the base gives \begin{equation}\label{fock:class-before-promotion} [L_{\lambda,d}]= \left[\bigoplus_{d(\mathbf r)=d} (\mu_{\mathbf r})_*\Q_{C_{\mathbf r}} \otimes\Th^{E(\mathbf r)}[2A(\mathbf r)]\right]. \end{equation} Each box on the right has weight $e_b+2(a_b+1)=h_b$. Hence both sides of \eqref{fock:class-before-promotion} are pure of weight $d$. The component of weight $d+k$ in their classes is exactly $(-1)^k[\pH^k]$. Semisimplicity identifies the corresponding perverse objects. A term on the right has perverse degree $R(\mathbf r)-2A(\mathbf r)$ because $\mu_{\mathbf r}$ is finite and $C_{\mathbf r}$ is smooth of dimension $R(\mathbf r)$. This proves Theorem~\ref{fock:relative-theorem}. Twisting by the symmetric charge $\Th^{-d}$ changes the box generator to \begin{equation}\label{fock:ordinary-charge} v^{h_b}(\delta_{h_b})_*\Q_C(a_b+1)[2(a_b+1)]. \end{equation} Thus the constituents of $\Th^{-d}L_{\lambda,d}$ lie in the ordinary Hodge-module image, with trivial auxiliary monodromy. This conclusion retains the $H^1(C)$ and all its exterior powers under proper integration. \subsection{Properness of the positive spectral map} The positive spectral map $P_{r,\chi}(X)\to\cA_r^L$ records the characteristic polynomial of $az+b\theta$. We will use its properness both for the comparison endpoint and for a closed filtration of the whole nil locus. \begin{proposition}\label{fock:properness} The positive spectral maps on $P_{r,\chi}(X_0)$ and $P_{r,\chi}(X_1)$ are proper. The restriction of the latter to $N_{r,\chi,\red}$ is equivariant for \eqref{fock:antidiagonal} and the positive grading on $\cA_r^L$, and admits an $H$-linearized ample line bundle. \end{proposition} \begin{proof} The closed immersions $f_s$ of \eqref{source:scalar-base-map} and the finite spectral addition of Lemma~\ref{source:finite-addition} show that, for each rank partition, the map \begin{equation}\label{fock:cycle-cover} \prod_i(\cA_{r_i}^{K_C}\times\A^1)\longrightarrow\cA_r^L \end{equation} is finite. On every compact irreducible curve in $X$, the function $z$ is constant, and its projection to $\Tot(K_C)$ is a spectral divisor finite over $C$. Thus the finitely many maps \eqref{fock:cycle-cover} parameterize all compact effective cycles of rank $r$, with their multiplicities. Given a DVR family in $\cA_r^L$ and a generic stable pair, after a finite extension its generic cycle lifts to one of these finite maps. Its lift extends over the integral-closure DVR and has a limiting cycle entirely in $X$. The projective stable-pair moduli on $Y=\mathbf P_C(\OO_C\oplus K_C\oplus\OO_C)$ extend the generic pair. Separatedness of its projective cycle identifies the limit with the cycle just constructed. Purity of the limiting sheaf prevents an additional isolated support component at infinity. The extended pair is therefore supported in $X$. Boundary avoidance descends through the finite extension, proving properness. For $X_0=\Tot_C(L\oplus A)$, the pullback of $A$ to the normalization of any compact irreducible curve has negative degree. Its transverse coordinate is consequently zero. Every compact cycle lies in $\Tot_C(L)$ and is a finite spectral divisor. The same DVR argument, now with the ordinary positive spectral coefficients as the cycle parameters, proves properness for this row as well. At a geometric nil point, the commuting $\theta,z$ are simultaneously triangularizable over the curve's generic field after extension, and the diagonal entries of $z$ vanish. Hence \begin{equation}\label{fock:nil-equivariance} \operatorname{char}(az+b\theta)=\operatorname{char}(b\theta),\quad \operatorname{char}(au^{-1}z+bu\theta)=\operatorname{char}(bu\theta). \end{equation} These are identities of regular functions on the reduced nil locus, and give the asserted positive-base equivariance. The $H$ action extends to $Y$. An $H$-linearized polarization induces linearizations on the projective section factor and the Pl\"ucker line in the stable-pair GIT construction \cite{ST}. It commutes with the framing linear group. Passing to the invariant graded ring and a divisible Veronese gives a linearized ample line on the projective stable-pair compactification. Its restriction gives the required line on the nil locus. \end{proof} \subsection{The whole nil locus} Proposition~\ref{fock:properness} and Lemma~\ref{fock:closed-filtration} give a finite closed filtration of $N_{r,\chi,\red}$ by whole attracting schemes over unions of complete fixed components. These are the reductions of the ambient attractors. Indeed, on an arbitrary ordinary ring the $j$th characteristic coefficient of $z$ in an attracting family has Laurent form $u^{-j}c_j$. Extension through zero forces $c_j=0$, by injectivity of $R[u]\to R[u,u^{-1}]$. On a reduced test base the orbit extension is reduced and therefore factors through $N_{r,\chi,\red}$; the converse is immediate. The universal reduced attractors and their limit maps consequently agree. Hodge modules and their six operations are insensitive to nilpotent thickenings, so \eqref{fock:compact-localization} computes the layers with exactly the coefficient \eqref{source:nil-coefficient}. The baseline Euler characteristic and $H$-index are \begin{align} \chi_0(\lambda) &=\chi\!\left(\bigoplus_{(i,j)\in\lambda}K_C^{-j}\right) =-h\sum_{(i,j)\in\lambda}(2j+1) =-h\norm{\lambda}^2,\label{fock:baseline-chi}\\ I_0(\lambda) &=\sum_{b\in\lambda}\chi(K_C^{a_b+1}) =h\sum_{b\in\lambda}(2a_b+1) =h\norm{\lambda}^2=-\chi_0(\lambda). \label{fock:baseline-index} \end{align} The positive transverse hook line is $K_C^{a_b+1}$, explaining the second formula. For fixed $\lambda$, the defect is $d=\chi-\chi_0(\lambda)$. The corresponding complete attracting layer has compact cohomology \begin{align} &R\Gamma(D_{\lambda,d,a},\Phi_{d,a}\otimes\La^{\chi+I_0+a}) =R\Gamma(D_{\lambda,d,a},\Phi_{d,a}\otimes\La^{d+a}), \nonumber\\ &\hspace{22mm}\bigoplus_aR\Gamma(D_{\lambda,d,a}, \Phi_{d,a}\otimes\La^{d+a}) =R\Gamma(\Sym^dC,\Th^d[-2d]L_{\lambda,d}). \label{fock:whole-layer} \end{align} The latter is pure of complex weight zero by \eqref{fock:relative-purity}. The symmetric charge changes each generator of \eqref{fock:relative-series} to \begin{equation}\label{fock:normalized-box} \Th^{h_b+e_b}[2(a_b+1)-2h_b] =\TT^{l_b}[-2l_b]. \end{equation} Hence the perverse cohomology and the proper cohomology of the normalized layer are ordinary pure Hodge objects, with summands $(\mu_{\mathbf r})_*\Q_{C_{\mathbf r}}(-L(\mathbf r))[-2L(\mathbf r)]$. For fixed $r,\chi$ there are finitely many partitions, and each partition fixes $d$. Each remaining projective Quot scheme has finitely many connected components; its index and the weight of the linearized ample line are locally constant. Every filtration layer therefore selects an open-and-closed union of complete components. Its proper cohomology is a direct summand of \eqref{fock:whole-layer}, so has the same purity and ordinary coefficient properties. In the compact-cohomology sequences of successive closed filtration steps, the boundary maps join weights differing by one and vanish. The remaining extensions of a fixed weight split in the pure polarizable heart. This gives \eqref{fock:nil-cohomology}, including its degree and auxiliary monodromy assertions, and proves Theorem~\ref{fock:nil-theorem}. \begin{remark}\label{fock:grading-distinction} The perverse degree in \eqref{fock:perverse-formula} belongs to the defect support map to $\Sym^d C$. The Hitchin perverse degree is extracted from the source primitive using $x$ in Section~\ref{sec:recovery}. Theorem~\ref{fock:nil-theorem} supplies the ordinary Hodge cohomology and its purity in each degree. \end{remark} \section{Hodge recovery and the character-variety formula} \label{sec:recovery} We combine the two calculations before taking any numerical realization. All formal variables in this section are even. For a bounded Hodge complex $K$, its class is $[K]=\sum_j(-1)^j[H^j(K)]$. Let \begin{equation}\label{rec:curve-zeta} \zeta_C(y)=\sum_{m\geq0}[H^*(\Sym^mC)]\,y^m =\frac{\bigwedge_{-y}H^1(C)}{(1-y)(1-\TT y)}. \end{equation} The last identity is the symmetric-product formula for a curve \cite{Macdonald}; it is an identity of Hodge classes, with the derived symmetric action on $H^1(C)$ giving the exterior numerator. \subsection{The identity in full Hodge classes} Define the series of the original restricted physical coefficients by \begin{equation}\label{rec:nil-series-definition} Z_{\nil}(Q,x)=1+\sum_{r\geq1}\sum_\chi Q^rx^\chi[R\Gamma_c(N_{r,\chi},K_{r,\chi})]. \end{equation} Theorem~\ref{fock:nil-theorem} gives \begin{equation}\label{rec:nil-fock-series} Z_{\nil}(Q,x)=\sum_\lambda Q^{|\lambda|} x^{-h\norm{\lambda}^2} \prod_{b\in\lambda}\zeta_C(x^{h_b}\TT^{l_b}). \end{equation} The degreewise ordinariness and purity proved above place these classes in $K_0(\MHS)$. Choose an integer $d$ coprime to $r$ with $\zeta=\exp(2\pi i d/r)$, and set $\chi_*=d-r(g-1)$. Let $M_K$ be the smooth moduli space of stable $K_C$-Higgs bundles of rank $r$ and degree $d$, equivalently of Euler characteristic $\chi_*$. This is the degree matching the fixed twist in \eqref{eq:character-variety}. The source formula permits this choice of its coprime parameter. Let $h_K:M_K\to\cA_r^{K_C}$ be its Hitchin map, and put \begin{equation}\label{rec:hitchin-perverse-objects} P_i^K=\pH^i Rh_{K,*}\Q_{M_K}[2N_r],\qquad A_i=[R\Gamma_c(\cA_r^{K_C},P_i^K)]. \end{equation} The scalar primitive of Section~\ref{sec:scalar}, restricted to the characteristic-nil locus, is \begin{equation}\label{rec:scalar-primitive} x^{i+1+u+v}P_i^K[-i-2v]\otimes\TT^{v-N_r}, \qquad u,v\geq0. \end{equation} Proper base change and compact integration of the finite spectral addition maps preserve its symmetric-power operations. Taking the plethystic logarithm of the source identity, summing $u,v$, and using the shift in \eqref{rec:scalar-primitive} give \begin{equation}\label{rec:primitive-identity} [Q^r]\Log Z_{\nil}(Q,x) =\frac{x\TT^{-N_r}}{(1-x)(1-x\TT)} \sum_i(-x)^iA_i. \end{equation} Both sides are now ordinary. The exact left inverse to the ordinary embedding into the monodromic category gives an injection on their Grothendieck groups, so the source identity gives \eqref{rec:primitive-identity} in $K_0(\MHS)$. Equivalently, define \begin{equation}\label{rec:recovered-series} F_r(x)=x^{-1}\TT^{N_r}(1-x)(1-x\TT) [Q^r]\Log Z_{\nil}(Q,x). \end{equation} Then \begin{equation}\label{rec:perverse-class-identity} F_r(x)=\sum_i(-x)^i [R\Gamma_c(\cA_r^{K_C},P_i^K)]. \end{equation} The variable $x$ records Euler characteristic; the source primitive identifies its exponent $i$ with the Hitchin perverse coefficient in \eqref{rec:perverse-class-identity}. All these operations are coefficientwise finite. For the finite rank range chosen at the start, the positive-source coefficients and the stable-pair endpoints vanish below $a_r$. For $r=\sum r_i$, \begin{equation}\label{rec:lower-bound-addition} \sum_i a_{r_i}-a_r=\ell\sum_{i2N_r$ \cite[Section 4.1]{BBD}. The ordinary weight bound for smooth varieties \cite[Theorem 2.1.2(5)]{HRV} gives $2k\geq D$ whenever $\beta_{D,k}\ne0$. Both exponents in \eqref{rec:barh-betti} are nonnegative; replacing $w$ by $-w$ cancels the displayed sign. Relative hard Lefschetz for the proper Hitchin map gives \[ \beta_{D,k}=\beta_{D+2(N_r-k),\,2N_r-k}. \] It interchanges the two exponents in \eqref{rec:barh-betti} and preserves the parity of $D$. \end{proof} \subsection{Ranks one and two} In rank one put $m=\chi+g-1$. For $m\geq0$, a stable pair is $\OO_C\to\OO_C(D)$ for an effective divisor $D$ of degree $m$, together with a field in $H^0(C,K_C)$ and a scalar. Hence \[ P_{1,\chi}(X_1)=\A^{g+1}\times\Sym^m C,\qquad N_{1,\chi}=\A^g\times\Sym^m C. \] The smooth physical coefficient is $\Q\otimes\La^{-(g+1+m)}$. Its normalization by $\La^\chi$ is $\TT^{-g}[2g]$. Compact integration over $\A^g$ contributes $\TT^g[-2g]$, so \[ R\Gamma_c(N_{1,\chi},K_{1,\chi}) \simeq R\Gamma(\Sym^m C,\Q). \] Consequently \[ [Q]Z_{\nil}=x^{-h}\zeta_C(x),\qquad F_1(x)=x^{-g}\TT^g\bigwedge_{-x}H^1(C), \] and the final substitution gives $(1+qt)^{2g}$. For rank two, let $Z_j=[Q^j]Z_{\nil}$. The partitions $(2)$ and $(1,1)$ give \[ Z_1=x^{-h}\zeta_C(x),\qquad Z_2=\zeta_C(x)\left(x^{-4h}\zeta_C(x^2) +x^{-2h}\zeta_C(x^2\TT)\right). \] Expanding \eqref{eq:plethystic-log} through rank two yields \[ [Q^2]\Log Z_{\nil} =Z_2-\frac12Z_1^2-\frac12\psi_2(Z_1). \] Under the weight character, \[ \psi_2(\zeta_C(x;s)) =\frac{(1-s^2x^2)^{2g}}{(1-x^2)(1-s^4x^2)}. \] The two partition terms and the two subtractions therefore give \begin{align*} \frac{H(\cM_{g,2};q,t)}{(1+qt)^{2g}} ={}&\frac{(1+q^2t^3)^{2g}}{(q^2t^2-1)(q^2t^4-1)} +\frac{q^{2g-2}t^{4g-4}(1+q^2t)^{2g}} {(q^2-1)(q^2t^2-1)}\\ &-\frac{q^{2g-2}t^{4g-4}(1+qt)^{2g}} {2(qt^2-1)(q-1)} -\frac{q^{2g-2}t^{4g-4}(qt-1)^{2g}} {2(q+1)(qt^2+1)}. \end{align*} This is the rank-two formula of \cite[Theorem 1.1.3]{HRV}. 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